No, it doesn't. The rent growth has been mostly stable over the last 2 decades and generally followed the inflation: https://fred.stlouisfed.org/graph/?g=1ZPQy And it increased only slightly recently (attributable to increased density as Houston relaxed the minimum lot requirements in 2014, btw).

Here is the REAL horror show: https://alfred.stlouisfed.org/graph/?g=1ZPRl

Houston is still the second most affordable large metro area in the US. And the per capita square footage is about two times that in NYC.