This seems like something that should be true, but I think I remember seeing a Mythbusters episode where they decided it didn’t make a difference. That show was more about entertainment than science, so I wondered if there was a more rigorous study done? I’ve definitely seen splashes of water(?) come out from flushes so that alone seems to argue for closing lids.
It’s written quite clearly on the door, but last time I used one I saved a tourist who would have had an unfortunate shower. She grabbed the door when I went out and was going in, I had to warn her and tell her to first let the door close so that the wash cycle would do its job.
You know what they say: you have to rush when Mother Nature calls. Under those circumstances, it can be difficult to read the instructions first. Good to know you saved the poor tourist a free chemical bath.