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 mean, we still CLEAN our toilets when they’re still just visibly stained with hard water or whatever causes rings and whatnot, so I can see the feeling better about being a huge component