Most public restrooms and locker rooms are open spaces that any person of any gender could enter at will if they wanted. They're not fortresses that a predator needs to disguise themself as the opposite gender to infiltrate and do harm. A sign on the wall saying men or women wouldn't stop anyone from entering if that's really what they wanted to do.

So yeah, the idea that this is what transfem individuals (because for some reason they're the sole focus of this petition) are trying to do is frankly absurd. Your personal insecurities are your problem and yours alone, and shouldn't form the basis of any kind of legislation or policy.

As for trans individuals in men or womens sports, that's a separate issue entirely. Have a discussion about that if you must, but leave restrooms and locker rooms out of it.