My answer is yes, I’d probably do it, assuming the profit is large enough and the lie isn’t on a topic that I greatly care about the truth of. But if everyone did this, everyone will be worse off, since their bargaining advantages will cancel each other, but their false beliefs will continue to carry a cost. It’s like a prisoner’s dilemma game, except that it won’t be obvious who is cooperating and who is defecting, so cooperation (i.e. not self-modifying to believe strategic falsehoods) probably can’t be enforced by simple tit-for-tat.
We as a society will need to find a solution to this problem, if it ever becomes a serious one.