Rental units on their own don't matter from a market perspective. There is some sentimental value in owning your own house but apart from that, costs have to be equal or demand shifts until they are.
Private equity can only make a profit if they corner the market and hold back units until prices massively increase.
This is possible to some extend but at its core it is most likely illegal because it requires collusion or a monopoly. So if that's an option, then there is a bigger problem than just housing prices.
But even in an entirely corrupt market, private equity cannot beat the market after Covid. Home office has become fully acceptable so huge fractions of the population are not bound to a specific city. People are free to choose where they live. There will be some municipalities that offer an abundance of housing so that prices will stay down.
Then, once people move there, companies will be founded there or move there, too. Wages will be lower because costs are lower, which will be a huge problem for the companies that didn't move.
So if the established municipalities allow the private equity companies to profit, they risk their own demise.
However, this depends on the willingness of the people to move. It is also contradicted by silicon valley being able to maintain its large housing prices.
My guess is that prices will rise for a while but some Asian city will crack the code and offer cheap housing despite being a cultural center which would make it the technological capital of the world.
I don’t have any proof of this but I strongly believe private equity, especially the big ones were the biggest beneficiaries of QE and they were closest to access the money.
If we are talking about a recession caused by fed tightening, PE won’t have any money. Especially as higher interest rates will mean they also can’t borrow lots to finance purchases or anything