Rates did not suddenly rise in the last month, they've been higher since April of 2022. A person signing a lease in April 2021 was in a much cheaper housing market than a person in April of 2022, which puts pressure on rents. The change in interest rates work in proportion, so going from a 2.75% interest rate to a 4% interest rate is more impactful than going from a 5% interest rate to a 6.25% interest rate even though the absolute rate change is the same. In April of 2021 interest rates were ~2.75% to April 2022 at 5% (an 80% increase in interest paid), April of 2022 to April of 2023 interest rates rose to 6.25% (25% increase in interest paid compared to the year prior, a smaller increase in absolute and substantially smaller in proportion). That is why the rental inflation rate peaked in March and has been falling since despite higher mortgage rates. In order for shelter to continue to rise, you need interest rates to continue to rise or housing price increases to accelerate. They won't. Not substantially, not in the same proportion that they had previously.
https://tradingeconomics.com/united-states/rent-inflationFeel free to bookmark this discussion. MoM Core PCE was .6% in August 2022 and .5% in September 2022. It would require a rapid increase in the inflation rate in August and September 2023 in order for the annual rate to maintain, much less rise. There is no indication of that happening right now. Prices are high, that doesn't mean that they are going to continue to trend higher at a rapid rate.
https://tradingeconomics.com/united-states/core-pce-price-index-mom .