I am councilling exercising caution before broadcasting downgraded estimations of personal competence - in the hope of avoiding failures caused by the planning fallacy. This could very easily be one of the cases where evolution is smarter than you are.

If the finding that "when people made their predictions anonymously, they do not show the optimistic bias" is correct then, this isn't really much of a "fallacy" in the first place. It would then be more of a signalling strategy - broadly similar to putting in low dollar initial estimates in the hope of getting hired.