The variance conjecture for isotropic log-concave vectors

Let $X$ be an isotropic log-concave random vector in $\mathbb R^n$, and let $|X|$ denote its Euclidean norm. The variance conjecture. There exists an absolute constant $C$ such that

$$\operatorname{Var}|X|^2\leq C\mathbb E|X|^2=Cn.$$

This conjecture concerns concentration of the Euclidean norm of isotropic log-concave vectors and is connected with the central limit problem for isotropic convex bodies. It is equivalent to the thin-shell width conjecture and is stronger than the hyperplane conjecture.