I want to see graphs that support this assertion. Yes, I expect that in an area with higher income housing becomes more expensive, but so does everything else. Unless you can show data to prove this I really doubt that an increase of median income of X% results in rent increasing X%, I rather suspect the rents increase to some smaller percentage of that together with increase in other things.

Rents are demand and offer, if the offer is perfectly static (no new housing/rental units are being made available) then if you have immigration in an area you should see rental pricing increasing according to that, rather than increase according to medium income.

EDIT: As for solving this with public housing, that'll simply make it that the market price is not reflecting the demand, but it doesn't change the underlying issue that there is more demand than offer. Which means that there will be other ways to "fight" for those limited units (instead of market price it will be "favors" or lottery or "first come first served" which tends to favor existing residents over newer ones).