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.
I saw the same or a very similar video.
Someone once tried to argue against it by saying it still got the droplets in the air with the lid closed so there’s no point. My counterargument was that it still contained a lot of the droplets by closing it and that it’s the most minor of inconveniences to close it so you should just do it anyways.