[edit] No supply + high demand = prices go up. Higher interest rates only help at the margins because it doesn't actually change the demand factor which is that the humans in America need a place to live, and the more they make, the more they can afford to pay. And it certainly doesn't change the supply factor!
[1] https://en.wikipedia.org/wiki/San_Francisco_housing_shortage
[2] https://www.cnbc.com/2021/09/14/america-is-short-more-than-5...