Depends on what's in the announcement. If it included a flat map with no gaps, correct distances, and a scale; a plausible explanation for eclipses, sextant, equatorial mount, star trails, 24 light in Antarctica in the winter; explanation of how they discovered it and why announce it now; an explanation of why the experiments and diagrams in science classes logically established a 99.999999% probability (aka true) - then I would believe them, or a FE. As clear and consistent as a schoolbook explaining seasons, eclipses, 24 hour sun at poles, etc.
And I have asked for an explanation of these things on FES, sextant/north star, map with a scale (NEVER have seen FE map with scale, please point me at one, ain't in the FAQ), etc. Got no answer or nonsense.
On the other hand, if the announcement went "The earth is flat and it is the greatest earth ever, much bigger than the other planets, the greatest earth ever known. Any reports that earth is round is fake news", then I would not believe them, all I would know from that is who appointed the new head of NASA.