The variance conjecture for isotropic log-concave random vectors

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

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

$$\operatorname{Var}|X|_2^2\le C\,\mathbb{E}|X|_2^2,$$

or equivalently,

$$\left(\mathbb{E}|X|_2^4\right)^{1/4}\le\left(1+\frac{C}{n}\right)\left(\mathbb{E}|X|_2^2\right)^{1/2}.$$

The conjecture concerns fluctuations of the squared Euclidean norm when independence is replaced by isotropy and log-concavity. The surrounding text presents it as a consequence suggested by the KLS framework and does not state a general resolution.