I think rising rents here is still compatible with low inflation however for a few reasons.
1. Housing is a luxury and positional good that is relatively supply constrained, so over time it will increase in price faster than other consumption items.
2. While we all follow market-based measures of rent, actual rents paid goes up more slowly because some of us landlords are softies who do not raise rents to market prices. Others have long term leases or legal restrictions. I’d have to check to be sure, but I think CPI would follow rents paid more than asking prices for vacant units.
3. Some rent increases are not either real or nominal inflation but reflect recent improvements to the rental. There’s also a gradual replacement of the lowest end units and building of mostly high end rentals that skews rents upward but isn’t the result of apples to apples price increases.
4. Rents are on a long-term downtrend in very big part of the USA.