The Lovász–Vempala thin-shell variance conjecture

Let $K$ be an isotropic convex body, and let $X$ be a random point from $K$. Lovász–Vempala variance conjecture.

$$\operatorname{var}\bigl(\lVert X\rVert^2\bigr)=O(n).$$

Under either definition of isotropy used in the source, this conjecture would imply that some natural random walks are significantly faster for isotropic convex bodies. Its status is not specified in the supplied text.