I know I'm supposed to say no, but...yes I do believe it.

I have a proof, too long to be contained within the margins of this page, that when supply goes up, price goes down

Go for it. Econ 100 textbook bs about supply and demand are trivially dismissed by a casual examination of housing and rental prices in the US in gentrified areas. What happens when the real world intrudes on academic platitudes is rather straightforward: pricing for everything goes up and the folks who were struggling get pushed out. Nobody who couldn't afford housing in the market before the shift is served.