Homes are up in price because zoning rules preclude development of new houses sufficient to meet demand, creating an imbalance in supply vs. demand. Interest rates only shift that equilibrium. Concretely, Japan's monetary and interest rate policy has been almost the same as the US for decades however they haven't seen an increase in housing prices in nominal terms since 1990. They federalize zoning so councils can't preclude you from building safe and reasonable housing, and this allows supply to meet demand.
Housing is driving inflation in the US, not responding to it - remember, inflation is the measured drop in purchasing power calculated from prices. Zoning causes house prices to go up, which in turn means the purchasing power of the dollar is calculated to be lower.
Punitive zoning rules are inflationary.
[1] https://en.wikipedia.org/wiki/San_Francisco_housing_shortage