That's kind of cheating though, because you're still de facto raising real estate costs (higher interest rate -> higher mortgage payment on the same loan principal).
Changing other components of real estate costs doesn't destroy the proportionality with prices, it only changes the ratio. The same is true of property tax -- raise the tax rate and sale prices decline while rents increase, because rents are also proportional to every other component of overall housing costs, not just the initial principal amount.
It's like saying retail prices aren't proportional to wholesale prices because if you raise commercial rents then the ratio changes. They're still proportional. If you raise rents then retail prices may increase because the retailer has higher costs. If you then raise wholesale prices, the retail prices don't stay the same just because they've already increased, they increase even more.
Changing other components of real estate costs doesn't destroy the proportionality with prices, it only changes the ratio. The same is true of property tax -- raise the tax rate and sale prices decline while rents increase, because rents are also proportional to every other component of overall housing costs, not just the initial principal amount.
It's like saying retail prices aren't proportional to wholesale prices because if you raise commercial rents then the ratio changes. They're still proportional. If you raise rents then retail prices may increase because the retailer has higher costs. If you then raise wholesale prices, the retail prices don't stay the same just because they've already increased, they increase even more.