are there any large/popular "facts" that the society does not believe in?
The society is a big tent, with no uniform set of beliefs, making this somewhat of a trick question. Any popular fact one might offer in answer to your question will have a spectrum of acceptance/rejection among the FES. For example, many of the FES here reject gravity, but by no means all of them. Many hold to some variation of the Gleason map found in the wiki, but by no means all of them.
That said, I think it is fair to say that anyone who holds to a flat earth must reject both the commonly taught sizes and distances of objects in the solar system, and the moon landings along with space flight in general (as those simply cannot work as described, if the earth is flat and the moon and sun are both small and close)