I think David Rabahy's comments illustrate a problem with OP's postulated scenario.

Individuals don't want to hold something that is fixed in value. They would rather hold something that appreciates in value. And markets are merely collections of individuals. As such, market forces will not conspire to make all monies asymptotically approach stable value, as long a there is an alternative available that appreciates in value. If indeed market forces will drive all monies to approach some prototype. The market forces in such a scenario would conspire to make all monies approach that other value-increasing alternative.