Assuming a perfectly spherical market, yes, high rates should push down prices.

It doesn't. I'm in DC, and prices have basically stayed put give or take a few percentage points. It's obvious why: if you own a house, you're going to be very resistant to taking a 100k+ loss when selling it, especially when the place you might buy is also stuck at a high price and will require paying double the interest rate.

So in practice, housing costs just shot up 40% until further notice for anyone who can't stay put.