I think we're missing the point. Yes, those attempting to dampen support for HERO used the possibility of predators in the ladies' but, that's not how voters saw the bill.
The vast majority of cisgender people have nothing against our using the restroom. However, they are concerned that it would no longer be possible to eject a male predator from the ladies' because of HERO - There would be fear of a lawsuit.
This is a legitimate fear (as opposed to the idea of a rapist claiming to be transgender, which has never happened.) A business owner *will* think twice about ejecting a male presenting person from a women's restroom, and I actually *can* understand why that makes people afraid.