Tilt-stable log-Sobolev conjecture for log-concave measures

Let $\mu$ be a log-concave probability measure on $\mathbb R^n$. It is $\beta$-tilt-stable if, for every $h\in\mathbb R^n$, the covariance operator of the tilted measure $\tau_h\mu$ is bounded by $\beta^2 I_n$ in operator norm, where $\tau_h\mu$ is the normalized tilt of $\mu$ by $e^{h\cdot x}$. Tilt-stable log-Sobolev conjecture. If $\mu$ is a $\beta$-tilt-stable log-concave probability, then

$$\rho_{LS}(\mu)\lesssim\beta.$$

The source states this as a weaker consequence of the preceding log-Sobolev KLS conjecture, using that tilt-stability implies sub-Gaussianity up to a universal factor. Its general validity remains open.