Why would that affect anything? Reduced demand for house purchases should result in higher supply for rental housing as owners rent it out instead of sell.
Because reduced demand for home purchases presumably means increased demand for renting. People have to live one way or another.
Additionally the driver of rent is ultimately an arbitrage argument. If I can buy a home and rent it out for more than the prevailing mortgage costs then rents must drop and home prices must rise, otherwise people will buy more property to rent out. Likewise the opposite is true. Home prices are dropping, so rent should drop too. Except for the fact that leveraged home costs are rising due to interest rates rising quickly. The article mentions some regions are seeing lots of new supply on the market so they can definitely impact the dynamic - but I don’t think coastal cities are seeing any new supply to speak of, so I doubt seriously rents are dropping in those cities. I would expect as costs to buy go up, rents will also rise assuming supply stays constant.
I assume it sort of balances out, if properties don’t sell, they may go on the rental market. Another thing is that the renters pool has some elasticity built-in: renters may move with relatives or roommates/significant other.