This is far from clear to me, the obvious formulations of that would seem to have infinite convergence time to me.

You've given a pretty clear explanation of one aspect of the problem— though another one with common causes is costless simulation: e.g. some early large number of stake holders can go fork it off and with no cost produce a simulated chain which a new node cannot distinguish from the original. They can do this at any point, even after exiting the system, years later— so it doesn't seem possible that any inside-system incentives can prevent them from doing so.