The Kannan–Lovász–Simonovits conjecture for logconcave measures

Let $\pi$ be a logconcave probability measure on $\mathbb{R}^{n}$ with covariance matrix $\Sigma$. Write $\psi_{\mathsf{KLS}}(\pi)$ for its KLS constant, and let $\psi_{\mathsf{KLS}}(n)$ denote the supremum of this constant over isotropic logconcave measures in $\mathbb{R}^{n}$.

KLS conjecture. The KLS constant satisfies

$$\psi_{\mathsf{KLS}}(\pi)=\Theta(\|\Sigma\|^{1/2}),$$

where $\|\Sigma\|$ is the largest eigenvalue of $\Sigma$; equivalently,

$$\psi_{\mathsf{KLS}}(n)=\Theta(1).$$

This conjecture was motivated by sampling from convex bodies and implies the thin-shell conjecture, which in turn implies the slicing conjecture. It is open; the best bound stated in the source is $\psi_{\mathsf{KLS}}(n)\lesssim\sqrt{\log n}$, due to Klartag.