A restrictive Idaho law — temporarily blocked by a federal judge Monday night — has amplified a charged debate about who should be allowed to compete in women’s sports, as transgender athletes have become increasingly accepted on the playing field while still facing strong resistance from some competitors and lawmakers.

While scientific and societal views of sex and gender identity have changed significantly in recent decades, a vexing question persists regarding athletes who transition from male to female: how to balance inclusivity, competitive fairness and safety.

There are no uniform guidelines — in fact the existing rules that govern sports often conflict — to determine the eligibility of transgender women and girls (policy battles have so far primarily centered on regulating women’s sports). And there is scant research on elite transgender athletes to guide sports officials as they attempt to provide equitable access to sports while reconciling any residual physiological advantages that may carry on from puberty.

Dr. Eric Vilain, a geneticist specializing in sexual development who has advised the N.C.A.A. and the International Olympic Committee on policies for transgender athletes, said that sports leaders were confronted with “two almost irreconcilable positions” in setting eligibility standards — one relying on an athlete’s declared gender and the other on biological litmus tests.

Politics, too, have entered the debate in a divided United States. While transgender people have broadly been more accepted across the country, the Trump administration and some states have sought to roll back protections for transgender people in health care, the military and other areas of civil rights, fueling a rise in hate crimes, according to the Human Rights Campaign.

In March, Idaho became the first state to bar transgender girls and women from participating in women’s sports.

The law, enacted in July by a Republican-controlled legislature with no Democratic support, required athletes to participate in sports based on their sex assigned at birth. The law mandated that all participants, including transgender athletes, answer a form about their sex, surgical procedures, medications and even whether they have had organs, like testes, removed. Any dispute in an athlete’s eligibility required a physical, genetic or hormonal exam conducted by a physician.

The ban was challenged by a transgender athlete in federal court in Idaho, claiming it violated equal protection guaranteed by the 14th Amendment to the Constitution. Chief U.S. District Judge David C. Nye temporarily halted the law on Monday, writing in an 87-page injunction that a “categorical bar to girls and women who are transgender stands in stark contrast to the policies of elite athletic bodies that regulate sports both nationally and globally,” which permit transgender women to participate in women’s sports in college and the Olympics under certain conditions.

While the ruling was not final, it was a victory for Lindsay Hecox, who is transgender and challenged the law in April, seeking to become eligible for the women’s cross-country team at Boise State University. “I’m a girl and the right team for me is the girl’s team,” Hecox said Monday in a statement. “It’s time courts recognize that and I am so glad that the court’s ruling does.”

The injunction was a setback for the Trump administration and for Barbara Ehardt, the Republican Idaho state representative who sponsored the bill. On Monday, Ehardt said she would still “stand strongly” behind her position.

In a recent interview, Ehardt, who played basketball in college and formerly coached a N.C.A.A. Division I women’s team, expressed concern that residual physical advantages that transgender athletes might possess could reduce the participation of cisgender women in sports. (Cisgender means their gender identity matches their assigned sex at birth.)

Ehardt said: “The progress that we, as women, have made over the last 50 years will be for naught and we will be forced to be spectators in our own sports.”

The Idaho case and a lawsuit in Connecticut that challenges the eligibility of transgender high school athletes have raised complicated questions about equitable access to sports, human rights and athletic advantages. Scientists have long said there is no single biological factor that determines sex, and the sex assigned at birth is not considered the sole determinant of gender.

‘One group prioritizes inclusion. Another group says we want fairness and safety.’

There is little or no scientific research regarding the performance of elite transgender athletes, experts say. But some evidence suggests that residual strength and muscle mass advantages largely remain when people assigned as males at birth undergo testosterone suppression for a year.

Complicating matters further, medical and ethical questions have arisen about whether any women should be required to lower testosterone levels just to play sports. The United Nations has called required hormone suppression “unnecessary, humiliating and harmful.” And there continues to be vigorous debate about the extent to which testosterone provides a decisive advantage in athletic performance.

Guidelines regarding transgender athletes represent “sport’s unsolvable problem,” said Ross Tucker, a South African exercise physiologist who is helping World Rugby develop its eligibility rules.

He said it seemed impossible to balance the values of competitive fairness, inclusion and safety because they conflict. “Therefore, you have to prioritize them,” Tucker said. “That’s the problem. One group prioritizes inclusion. Another group says we want fairness and safety” on the playing field.

At puberty, male athletes generally gain physiological advantages for many sports, like a larger skeletal structure, greater muscle mass and strength, less body fat, greater bone density, larger hearts and greater oxygen-carrying capacity. As a result, men and women mostly compete in separate divisions. At issue for scientists and sports officials is how much testosterone-suppression regimens reduce those advantages.

Even if transgender athletes retain some competitive advantages, it does not necessarily mean that the advantages are unfair, because all top athletes possess some edge over their peers, said Vilain, the director of the Center for Genetic Medicine Research at Children’s National Hospital in Washington.

“It’s like saying Usain Bolt’s abilities are unfair because he wins by so much each time,” Vilain said.

Contrary to fears expressed by some, there has been no large-scale dominance of transgender athletes in women’s sports.

One former athlete who expressed such fears, the tennis great Martina Navratilova, was widely criticized and walked back her comments after writing last year in The Times of London that it was “insane” and “cheating” to allow transgender women to participate in women’s sports. She wrote without evidence that any man could “decide to be female,” take hormones and become victorious and then reverse course “and go back to making babies.”

Veronica Ivy, who was known as Rachel McKinnon when she became the first transgender woman to win a world masters track cycling title, accused Navratilova of being transphobic and having “an irrational fear of something that doesn’t happen.”

Olympic historians say that no athletes at the Winter or Summer Games identified themselves publicly as transgender when they competed. At least two announced that they were transgender sometime after competing, including Caitlyn Jenner.

In recent years, transgender athletes have become more widely embraced and successful in sports from high school and college track to international weight lifting. The Tokyo Olympics, postponed to 2021 because of the coronavirus pandemic, could feature transgender women such as the BMX freestyle rider Chelsea Wolfe of the United States, the volleyball player Tiffany Abreu of Brazil and the weight lifter Laurel Hubbard of New Zealand.

Connecticut decided to split high school athletes by gender identity, prompting a lawsuit.

Governing bodies have varied in their approaches toward transgender athletes. Connecticut, for example, permits athletes to compete according to their gender identity, without restriction.

Last February, the Connecticut Interscholastic Athletic Conference policy was challenged by three cisgender female athletes, Selina Soule, Chelsea Mitchell and Alanna Smith. The lawsuit, still being adjudicated, referred to two transgender sprinters, Terry Miller and Andraya Yearwood, who have won 15 state titles between them. Their dominance in winning races deprived cisgender athletes of athletic opportunities and potential college scholarships, the suit argued.

In May, the civil rights division of the United States Department of Education ruled that Connecticut’s high school policy violated Title IX, the federal law that prohibits discrimination on the basis of sex.

“All the biological females know who is going to win before we even start, and it’s sad to see that all our training just goes to waste,” Smith, one of the three plaintiffs in the lawsuit, said in an interview.

The Connecticut high school association’s position is that multiple federal courts and government agencies have acknowledged in reference to Title IX that the term “sex” is “ambiguous” and historical usage of the word “has not kept pace with contemporary science, advances in medical knowledge and societal norms.”

Miller, the star transgender sprinter, said in a statement in February, “The more we are told that we don’t belong and should be ashamed of who we are, the fewer opportunities we have to participate in sports.”

The N.C.A.A. welcomed transgender athletes beginning in 2011.

Today, about 200,000 athletes compete in women’s college sports. Joanna Harper, a researcher and medical physicist, estimated that about 50 are transgender.

Harper, who is transgender, said that different levels and types of sports should tailor policies to their unique circumstances, including testosterone-suppression requirements for top transgender athletes beginning in high school, but with an eye toward inclusivity.

“Where there’s no professional contracts, no money, no Olympic glory, we should be very inclusive about that,” Harper said.

The N.C.A.A. policy says that issues of basic fairness and equity “demand the expansion of our thinking about equal opportunity in sports.” The organization requires that transgender women undergo testosterone suppression treatment for a year before becoming eligible for women’s events. But the N.C.A.A. says it does not set permissible limits of testosterone for transgender athletes.

In August 2019, June Eastwood of the University of Montana became the first known transgender woman to compete in an N.C.A.A. Division I women’s cross-country race. Before transitioning, Eastwood had been a top runner for Montana’s men’s cross-country and track teams. But in women’s cross-country, she finished 60th at a regional meet and did not qualify for the national championships.

Four years earlier, Harper had published the first study on the effect of hormone therapy — testosterone suppression and estrogen — on the performance of transgender athletes.

Her research found that a nonelite group of eight transgender distance runners was no more competitive as women than as men. Her findings suggested that a performance advantage was not always maintained over cisgender women as transgender women faced a reduction in speed, strength, endurance and oxygen-carrying capacity.

Harper noted, though, that her research applied only to distance runners and that transgender sprinters may retain an advantage over cisgender sprinters because they tend to carry more muscle mass to propel themselves over shorter distances.

In May 2019, the first known transgender athlete won an N.C.A.A. track championship, when CeCe Telfer of Franklin Pierce University in New Hampshire took first place in the Division II women’s 400-meter hurdle race. Telfer’s transition seemed to follow, at least partially, Harper’s theory about sprinters.

In two seasons for the Franklin Pierce men’s team, Telfer did not qualify for the national collegiate championships, ranking no better than 200th in the hurdles event. On the women’s team, she ran slightly slower in winning the 400-meter hurdles (57.53 seconds) than she had before transitioning (57.34). But Telfer was faster after transitioning in the 60-meter sprint (7.63 seconds compared to 7.67) and the 400-meter sprint (54.41 seconds compared to 55.57).

Telfer attributed her success, in part, to a newfound motivation to compete, telling Outsports.com that she felt a release from gender dysphoria, which left her “uncomfortable in my skin.”

But Gregory A. Brown, a professor of exercise science at the University of Nebraska-Kearney, wrote in an expert declaration for the state of Idaho in its case that Telfer’s performance “provides some evidence that male-to-female transgender treatment does not negate the inherent athletic performance advantages” of a person who experienced puberty as a male.

Some experts say the science undermines any hope for universal guidelines.

A 2019 Swedish study of 11 transgender women found that, after a year of undergoing testosterone suppression, they experienced only a negligible decrease in strength in their thigh muscles and only a 5 percent loss of muscle mass.

One of the researchers, Tommy Lundberg of the Karolinska Institute outside Stockholm, said in an interview that the participants were untrained and that it was difficult to speculate about physical changes to elite athletes because “there are no longitudinal studies.” He added, “I’m not sure there will be any reliable data at any point.”

The International Olympic Committee allows transgender female athletes to compete in the Games if they reduce their serum testosterone levels below 10 nanomoles per liter for a year and maintain the lower levels during their careers. Transgender men can compete without restriction.

According to World Athletics, track and field’s governing body, the general testosterone range for cisgender women is .12 to 1.79 nanomoles per liter, compared with 7.7 to 29.4 nanomoles per liter for men after puberty. In other words, World Athletics says, the lowest level in the men’s range is four times greater than the highest level in the women’s range.

Separately, track and field has guidelines specifically for intersex athletes, competitors born with biological factors that don’t fit typical descriptions for males or females. Those who possess a rare chromosomal condition are required to reduce their testosterone levels even lower, to five nanomoles per liter, in races from the quarter mile to the mile.

One of the athletes affected is Caster Semenya of South Africa, the two-time Olympic champion at 800 meters, who was classified as female at birth, identifies as a woman and is challenging the track and field policy in an ongoing case.

The I.O.C. has been widely expected to require transgender Olympic athletes to adhere to the five nanomole limit after the Tokyo Games.

According to new research, which examines available studies of testosterone suppression, evidence shows that even a reduction to one nanomole per liter — squarely within the average female range — only minimally reduces the advantages of muscle mass and strength retained as men transition to women.

That undermines the attempt of sports organizations to set universal guidelines, said Lundberg, a co-author of the study, which is undergoing peer review. He recommends that individual sports set their own policies.

“It is easy to sympathize with arguments made on both sides,” Lundberg said of gender identity versus biology. But, he added, “It is going to be impossible to make everyone happy.”