Not literally forever, of course, but I think it's naive to say people's inability to have the lifestyle they want will deter increasing rents. Empirically, society on balance doesn't care one tiny bit how much hardship people endure, and people will endure the hardship. I'm not seeing what mechanism is assumed to counter that.
Increasing supply. Which in this case is difficult due to the alliance of existing property owners and useful idiot "anti-gentrification" protestors.