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.
No its about liquidity and utility, have you even read the paper? This is more than just simply "thinking" about how your perfect money would work in theory. This is based on game theoretical fundamentals with mathematical relevance and one which a certain high IQ society member spent the majority of his working life trying to figure out. Note that in all his talks he declined to ever comment on bitcoin almost purposefully, and nash was known to try to hide patterns within patterns for people to try to deduce true meanings of his words, but read the paper and it becomes crystal clear on what he is getting at.. well to me anyway.