when asset prices rise, it becomes more difficult to reach a certain level of real wealth.
This is the way most people experience the "wealth gap"--a decline in economic mobility. Stagnation possibly explains the rise of so-called 'populist' movements. Those stuck on the lower rungs of the economic ladder see no way up and seek unconventional solutions to their difficulty.
I'm no economist, and don't even understand economics very well, but your line struck me: "it becomes more difficult to reach a certain level of wealth".
To some people it seems not more difficult, but impossible.
Yes, unfortunately. And I think this impossibility "to reach a certain level of wealth" contributes to the discontent of many people adding fuel to the fire of Black Lives matter in the US and the Yellow Vest Movement in Europe.