Actually, there are several areas other than sports in which conflicts have arisen, and will arise, insofar as legal efforts to protect different groups of people from discrimination based on sex and gender are concerned.
Prisons, for example.
As I am sure you know, in many states in the US, adult male convicts are now placed in either men's or women's prisons based on their claimed gender identities. This hasn't worked out well for the female convicts these males are locked in with. Quite a few of the male convicts placed in women's facilities have raped, assaulted, menaced, intimidated and sexually pursued female inmates. In New Jersey, one male murderer impregnated two female inmates at the Edna Mahan women's prison pretty soon after arriving there. In Ilinois, the first male inmate transferred to a female prison based on gender identity claims was another murderer - and on the first day "she" was allowed in with the general pop, "she" raped a female inmate and is accused of raping a second female prisoner later on.
So whose rights come first here? The rights of male inmates to serve their time behind bars in female prisons if they say they "identify as" and "feel like" women themselves? Or the rights of female inmates not to be subjected to the cruel and unusual punishment of being locked up behind bars with male criminals who are twice their size and strength - and many of whom so far have been brutal murderers, rapists, child molesters and serial killers of women?
Whatever crimes some female inmates have committed, how is it constitutional, just and humane to lock women in cages with male convicts based on the males' claims about their self-perceptions and identities?
Other areas of conflict are communal toilets, changing rooms, showers, spas, dorm and bunk rooms, hospital wards, nursing home accommodations, shelters, rape refuges, in-home health care, barracks, medical services and research, support groups, crime statistics, etc.
I believe people of both sexes should be protected against discrimination based on our sex, sexual orientation, and gender, gender expression and gender identity (if we have one). But these are very thorny areas of law that are as-yet unsettled - and it's naive and a grave mistake to pretend that there aren't conflicts all over the place between rights and protections based on sex and rights and protections based on gender, gender identity and expression.
But generally speaking, I think it would be a huge step backwards if women's and girls's sports, women's locker rooms and loos, etc, and women's rights were redefined as sports, spaces and rights for "feminine" people rather than for female people.