I don't follow Anthropic closely enough to know what claims its CEO has made, but it is factual that Altman is a pathological liar. You can observe this for yourself by reading and listening to the things he says and then comparing them to reality. We have years of evidence to look back on. The chasm between Altman's reality and everyone else's is so large and so well-known that it was one of the chief factors cited by the board when he was fired.

(I would then argue that he was re-hired specifically because others involved with OpenAI understood that it is literally his job to lie and that OpenAI would not be where it is today as a corporate behemoth rather than a research non-profit without a world-class liar marketing it, but that is merely conjecture.)