The equilibrium price would be set by the demand for money-as-currency. But in the limit of instantaneous (electronic) settling of payments, this demand premium converges on zero, with the only residual demand being for money-as-store-of-value.

The purchasing power of hard money increases with time only to the extent that reality changes, i.e., for the same reason that each year you can buy more CPU cycles per inflation-adjusted dollar. And it's exactly as "deflationary" as the falling price of CPU cycles. This is perceived to be a problem only because of a double meaning of the word deflation: a contraction in the money supply causes a potentially harmful general drop in prices, whereas increased productivity causes a beneficial increase in the purchasing power of money. Those who fear the latter need to explain why it's bad that, e.g., smartphones keep getting better and cheaper every year.

Unfortunately, the usual strategy is to describe both kinds of price decreases as "deflationary", point out that deflation has historically been harmful, and then conclude that hard money must be harmful as well. That this logic is applied to hard money and not to smartphones is most parsimoniously explained by the political incentives that led to the abandonment of hard money in the first place, together with the associated displacement of descriptive "political economy" by prescriptive economics.