No they don’t, only a whack case would expect men to take care of their preference in a toilet seat.
If the feminists/media have created such a weird new concept, then men can just ignore the toilet seat entirely and let women put it up, or leave it down when they finish, as they prefer, it doesn’t much affect a man, either way.
Men can simply stay out of that whole new area of women’s craziness.
Problem solved... buy 2 toilets for the house if it’s affordable.