The variance conjecture for the Euclidean norm

Let $X$ be an isotropic log-concave random vector in $\mathbb{R}^n$, and let $|\cdot|_2$ denote the Euclidean norm.

Variance conjecture—bis. There exists a constant $C$ such that for every isotropic log-concave random vector $X$,

$$\operatorname{Var}|X|_2\le C.$$

This is a direct thin-shell formulation for the Euclidean norm and is stronger in form than merely asserting $\operatorname{Var}|X|_2^2=O(n)$. The source calls it another simple conjecture arising from the Poincaré viewpoint and gives no resolution.