https://www.bls.gov/cpi/factsheets/owners-equivalent-rent-an...
if rents rise, and other CPI items also rise, then it's not true that this is "all else being equal". If other items also rise, it means that there's more money being spent, which implies that people have more money to spend, and thus, income must be rising.
If rent rises are offset by costs of other items going down, then CPI won't rise - and this would happen if the rent is consuming more proportion of income, and thus, less is left available for the other CPI items.
I dont know which situation we are currently in today. I suspect that income is rising, and thus, inflation is high. Increasing rates would push up rent, but not push up income, and thus the proportion of rent vs income grows higher and thus lower inflation (in the future).
Or they will increase costs; which is (1) people (labor) demanding higher wages or (2) business increasing the costs of good to cover rent costs. Given that rent is already the biggest cost and has been growing steadily, it may be the case the other spending areas have been dialed to a minimum already.
how is higher rents fueling inflation? People paying higher rents (but on the same income) will have to forego some spending, thus reducing demand for some goods/services. So you'd actually expect that higher rents would drive _down_ inflation!