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.
Pretty sure aerosolized droplets from the toilets are a great way to spread diseases, especially digestion-related stuff
But yeah that’s not a scientific argument right there, just a hypothesis