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.
That assumes that it is worth avoiding a "fecal plume". I would think that useful research would tell us whether such a thing is actually bad for us, beyond sounding gross.
Now that you put it that way, let's take a moment of awareness that toilets in public restrooms rarely have lids, including toilets in restaurants and grocery stores 😱