and not opinion?

It's fairly unclear to me, I mean you have illustrations labeling the rooms generally (not much help) and then a reliance on common parlance and one's own definition of that as it pertains to sex vs gender.

What do say to the person that asserts with equal certainty that they are separated by sex and not gender?

For the record, I really could not care less about this issue. It could be gender anarchy in the bathroom for all I give a shit and I never turn a side eye at the person sneaking into the wrong bathroom for a shorter line.