Demand for a place to live is much more inelastic.
And rate hikes make it harder to purchase a home, by raising the cost of them. That forces out of the home-buying market those who would have bought a home at the pre-hike rate … and forces them back into the rental market. Seems to me like demand goes up.
(Certainly, it makes it harder for landlords to acquire new properties, but I think corporations are going to weather it better than people — i.e., people will be forced out of that market before corporations.)