Not really. In a system where demand is fairly inelastic (everyone needs somewhere to live and the only real flex is having roommates/living at home/homelessness or renting two apartments) and where the supply is currently extremely constrained, expenses are going to have next to no impact on rental prices.

For example, I was fortunately able to buy a townhouse two years ago (when interest rates were low) to live in. My mortgage is ~ $1,200/mo. Other units have been going on the rental market pretty consistently for ~$2,000/mo. Even with the increased interest rates, new landlord's would still have a net positive of ~$500/mo between the rent they receive and their mortgage payments. There might be a loss of profit, but with profits already so high, it's not going to affect rates on a macro scale.