You're right.

"...contractionary monetary policy increases both real and nominal rents. A 25 basis point unexpected increase in the 30-year fixed rate mortgage raises real (nominal) rents by 1.7 (1.4) percent 12-24 months following the monetary policy shock. We find that the effect is driven primarily by a shift in household demand from the owner-occupied market to the rental market. Using transaction level data, we show that when monetary policy tightens, renters are less likely to become homeowners and the home-ownership rate drops. This then crowds in the rental market and increases rents. We find that the increase in demand for rentals is accommodated by real-estate investors who capitalize on the higher rents by buying houses from owner-occupiers and renting them in out the rental market" [1].

Going back to the top, however, despite rent being "the single largest component of" CPI, we have not seen evidence anywhere of higher rates causing higher inflation. I suppose one could argue within the constant-unemployment state the effect of rates on inflation may be muted by the above. But again, being in an economy where private investment rivals personal consumption gives the Fed a lot more leeway on the upper limits of rate rises.

[1] https://business.columbia.edu/sites/default/files-efs/citati...