Log-Sobolev KLS conjecture

Let $\mu$ be a log-concave probability measure on $\mathbb R^n$, and let $\|f\|_{\psi_2(\mu)}$ denote the sub-Gaussian Orlicz norm defined by the best constant $C$ such that

$$\mu(|f|\geq r)\leq 2\exp\left(-\frac{r^2}{C^2}\right).$$

Let $D_n$ be the largest log-Sobolev constant among log-concave measures satisfying $\|x\cdot\theta\|_{\psi_2(\mu)}\leq 1$ for every direction $\theta$.

Log-Sobolev KLS conjecture.

$$D_n=O(1).$$

Equivalently, in the log-concave setting, the log-Sobolev constant should be bounded by the largest squared $\psi_2$-norm of a linear function. This is presented as the log-Sobolev analogue of the KLS conjecture; the source does not specify whether it has been resolved.