How do you "[render] the toilets genuinely gender neutral?"
It seems to me the men started using both restrooms, but the women refused to use cubicles that shared the same space as urinals? Why?
I imagine the most efficient gender neutral restroom scheme would be something like:
Upright urination has one dedicated queue. Sitting down urination has another queue. Defecation has another queue.
You could then have just one huge restroom with LEDs above cubicles that designate an opening for a particular queue like a parking garage. That way cubicles can be dynamically allocated based on the size of defecation vs urination queues.
Kind of like how at work you have 2 microwaves - one for people who need to zap their food for less than a minute, one for people that need to cook their lunch for more than a minute.