KLS conjecture for log-concave probability measures

Let $d\geq 1$, let $f$ be smooth on $\mathbb{R}^d$, and let $\mu$ be an absolutely continuous, log-concave probability measure on $\mathbb{R}^d$. Define

$$\lambda_\mu=\sup_{\theta\in\mathbb{S}^{d-1}}\operatorname{Var}_\mu[\langle\theta,X\rangle]=\|\operatorname{Cov}(\mu)\|_{\operatorname{op}},$$

where $\lambda_\mu$ is the largest eigenvalue of the covariance matrix

$$\operatorname{Cov}(\mu)=\mathopen{}\mathclose\bgroup\originalleft(\int_{\mathbb{R}^d}x_i x_j\,\mathrm{d}\mu(x)-\int_{\mathbb{R}^d}x_i\,\mathrm{d}\mu(x)\int_{\mathbb{R}^d}x_j\,\mathrm{d}\mu(x)\aftergroup\egroup\originalright)_{1\leq i,j\leq d}.$$

KLS conjecture. There exists a universal constant $\mathbf{C}_{\mathrm{univ}}>0$ such that

$$\operatorname{Var}_\mu f\leq \mathbf{C}_{\mathrm{univ}}\,\lambda_\mu\,\mathbb{E}_\mu\mathopen{}\mathclose\bgroup\originalleft[\|\nabla f\|_2^2\aftergroup\egroup\originalright].$$

This is a central open problem in convex geometry concerning dimension-free Poincaré inequalities for log-concave measures. The conjecture has been open for three decades and is relevant to high-dimensional sampling algorithms; the paper uses $\beta$-ensemble central limit theorems to provide a consistency check for $p$-Schatten balls.