Kristi Noem doesn't just want to 'preserve biological women's spaces'. If you don't accept that trans people are who they say they are then the only alternative is that they are mentally ill or perverts.
Lots of people on Mumsnet would agree with that, but I'm guessing that they're not going to believe that homosexuals are mentally ill or perverts. That's what people used to say about them, that's what the traditional view is, and that's what people like Kristi Noem believes.
Do you think that Kristi Noem would accept someone like Kathleen Stock wearing men's clothes, having short hair and no makeup? She wouldn't accept it any more than she would accept men wearing women's clothing, no matter how much Kathleen Stock might insist there's nothing sexual about it or that she's not trying to be like a man.
Do you think that Kristi Noem would accept Kathleen Stock's marriage to another woman? She is against gay marriage. She isn't going to believe that Stock's lesbianism isn't a paraphilia.