Does a State have the right to bar transgendered individuals from using public restrooms which they want to use?

Question

Answer 1

It shouldn't

Answer 2

If I'm a therian, and think I am a cat, should schools and other places with public restrooms be required to provide me with a litter box?

Answer 3

I don't think a State should, however, I think a future US Supreme Court case could decide the issue.