That's what everyone keeps saying. INAL but that would give any fired executive hella leverage when the board finally decide to unveil what they've been hiding. I mean, they'll go to discovery and ... perjure themselves? Really?
What claim would the executive have? They can be fired by the board for any reason (except for a few explicitly prohibited reasons), so "they lied about why they fired me" doesn't work as a claim unless you're alleging the real reason is because they're racist or something similar.