The weak and strong moments conjecture for log-concave random vectors

Let $X$ be a log-concave random vector and let $\|\cdot\|$ be any norm. Its dual norm is defined by

$$\|z\|_{\star}=\sup_{\|x\|\le1}|\langle z,x\rangle|.$$

Weak and strong moments conjecture. There exist $C,c>0$ such that for any log-concave random vector $X$ and any norm $\|\cdot\|$,

$$\left(\mathbb{E}\|X\|^p\right)^{1/p}\le C\,\mathbb{E}\|X\|+c\sup_{\|z\|_{\star}\le1}\left(\mathbb{E}\langle z,X\rangle^p\right)^{1/p}.$$

This proposes a universal comparison between strong moments of an arbitrary norm and one-dimensional weak moments. The source attributes the question to Latała and gives Gaussian and Rademacher analogues, but does not state a general resolution.