All the locker room stuff aside, I think there's legitimate concern here. From a physiological standpoint men are built differently than women and I don't know that any specific hormone therapy can level that playing field, and anytime a transgendered person were to win it would always be questioned.
Keep in mind those physiological differences come as a result of development driven by.... drumroll please... hormones. One of the few studies on the effect of hormonal transition unfortunately has a small sample size, but by using US military physical performance measurements they can at least have some sort of starting point.
https://bjsm.bmj.com/content/55/11/577
The easy bit that shows up here is that testosterone works fast. Trans men caught up with their cis peers right quick. Trans women did show a shrinking edge during the two year period that the study looked at. But there's a couple wrinkles here. First is that the sample size is small, leaving rather large error bars in the measurements, where both sample groups have error bars that overlap the control groups. Second, is the short window for the study, as there is a
trend without a clear endpoint to the trend. A longer study is needed to determine where these people ultimately land in the long term. Third, when treatment starts also greatly determines the outcome, many of the physiological changes relevant here happen during puberty. Prior to puberty, outside of gonadal development differences in the womb, the physiological differences are quite muted.
So here's the thing, I'm left with a couple big concerns here. First is that forcing trans people to compete strictly on "what gonads we saw at birth" basis still isn't fair to women's sports if we account for the clear advantage testosterone confers in many sports. Nor is putting all trans people in one bucket, in a world were women's teams who perform consistently better in professional competition get underpaid, and less attention (which makes widespread concerns of "what about women's sports" ring a bit hollow). Second is that this underscores the other prong of the attack on trans rights to block the ability of trans people to access puberty blockers so they have time to make a more reasoned decision about which puberty they want. An approach that makes much of what the study above looked at moot for many, as more trans people would develop physiology more in line with the group they are transitioning to.
Ultimately, this is just one facet of a multi-faceted approach to restrict how trans people can exist in society. If you transition at all, you can't do X and Y. And if you transition as an adult, you can't do Z either. And you can't transition during puberty, so you still can't do Z. It's a double-bind, and of course trans people are going to be sick of the paternalistic nature of it all.
And this all I'll say on the subject.
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