The marketcap is defined by the price and the circulating supply. I think most Austrian economists believe that an increase in supply of a currency creates a downward pressure on the unit price of a currency. I agree with this and I believe you do too. I don't like to use the term "inflation" because not everyone agrees what the word means.
The downward pressure doesn't immediately affect the value though. I think it depends on how much transfers are taking place. If there is little no economic activity, the price will take incredibly long to get re-discovered. In fiat the economic activity is much higher than that of any blockchain, so the incrase of supply has a relatively quick effect on fiat denominated prices.