It's often easier to think about the reverse situation -- imagine there's an error at the bank and the property owner's mortgage goes poof and his cost of operating the rental is cut in half -- do you think he'd lower the rent for his tenants? Of course not.

It's not a perfect market, rents are sticky, landlords don't profit maximize all of the time (e.g. they might keep the rent the same for several years for good tenants), but in general landlords will charge the highest amount they can charge at all times regardless of what their cost basis looks like.