Under self-ID as the TRA's want it, you cannot lie about this. Whatever you say is axiomatically true. It is true because you said it.
Yup. There is no "self-ID, but". Once you introduce the "but", self-ID collapses.
If policies, both in the private and public sphere, need to distinguish between men and women (regardless of whether these concepts are determined via sex or gender), then self-ID is very shaky ground. In this scenario, it creates incentives the moment people see the possibility entirely in their hands of being granted access to that which is limited or denied on the basis of sex/gender differentiation.
On the other hand, if there is no need for public and private policies to make the distinction in the first place, then self-ID becomes as trivial as "nice hat!" Under this scenario, of course, everything goes and no objections from me.
The thing is, I'm of the idea that we should make the least distinctions possible between men and women when creating policies, but that some distinctions are inevitable if we want to avoid a greater harm than the good "absolute equality" produces.
Self-ID can be cool most of the time, though. Like, again, "nice hat!" I'm all in for that.