This is wrong. The article you cited has data up until August 2023.

Asking rent has been declining since September 2023: https://www.rent.com/research/average-rent-price-report/

Bureau of Labor Statistics has a 12-month lag when it comes to housing price, which means that elevated rent is still counted in the CPI print. Expect to see CPI continues to fall over the next few months. The current round of inflation is over.

All the right wing economists who predicted a scary recession throughout 2022 and 2023 were wrong.