The weak thin shell conjecture for log-concave random vectors

Let $X$ be a log-concave random vector, let $|\cdot|_2$ denote the Euclidean norm, and define its weak $p$-moment by

$$\sigma_p(X)=\sup_{|z|_2\le1}\left(\mathbb{E}|\langle z,X\rangle|^p\right)^{1/p}.$$

Weak thin shell conjecture. There exists $c>0$ such that for every log-concave random vector $X$,

$$\forall\,p\ge1,\qquad \left(\mathbb{E}|X|_2^p\right)^{1/p}\le\mathbb{E}|X|_2+c\,\sigma_p(X).$$

This seeks to remove the multiplicative constant in the known strong-versus-weak moment estimate for log-concave vectors. It is presented as a weaker consequence suggested by the thin shell conjecture; no resolution is given.