In addition to the current supply constraint keeping prices high, mortgages being fixed rate means higher interest rates don't really drive prices down (although lower interest rates do drive prices up). If you have a mortgage for $600k at 3% interest rate and rates go to 6.6% (like they are now), the naive loan math says your house is now worth $400k. Who has $200k in equity built up in their home and is willing to walk away from it? Virtually no one, so prices stay high until people get desperate to sell.
This leads to the fun conclusion that raising interest rates might actually make inflation worse. Rent/rent equivalent is already 30% of CPI, so increases in housing costs have a big effect on overall inflation.
This leads to the fun conclusion that raising interest rates might actually make inflation worse. Rent/rent equivalent is already 30% of CPI, so increases in housing costs have a big effect on overall inflation.