The Flat Earthers claim that all of the stars, planets, galaxies, etc are hovering 3,000 miles above our heads and that they all dance around in complicated patterns to try their best to mimic what would happen if the Earth was round and they were a bazillion miles away.
Since ZERO professional astronomers agree with that - and they have bucketloads of evidence to show otherwise (eg redshift) - they pretty much have to be lying and therefore must be lumped into the big "NASA conspiracy" bucket.
Adding a few gravity wave detector people to the conspiracy is neither here nor there in the grand scheme of FE paranoia.
So there is not much chance of the FE'ers really being convinced by the small additional evidence.