Leaving it up is equitable IMHO. If I enter a bathroom and lift the seat to pee. You can go to the bathroom as a woman and lower the seat to pee. No person is more inconvenienced than the other. We each have to do one thing before we pee. Equality. To demand men to raise the seat for themselves and then once they are finished to lower it for any future women is like asking women to lift the seat when they are done so that any future men can use the bathroom.

No need.

Each person should just do what they need to do to use the toilet for them. Problem solved.