I'm pretty sure that this is correct.

More precisely, I'm pretty sure that one simply doesn't want to have ANY reputation regarding trustworthyness, truth, or whatever. Make issues of truth salient and you loose. Even a reputation for always communicating honestly (no efforts at deception) costs you status because it makes you a less valuable ally, less capable of desirable forms of partiality, and above all, weird. Being seen as a trusted neutral third party is at best a weak consolation prize, and one that is only possible if you are also seen as either a) not having your own agenda, or b) not having an agenda that anyone is allowed to question.

By contrast, politicians who are caught in lies repeatedly pick themselves up and go back to being high status politicians after wiping the dirt off their faces.