Latała's moment conjecture for norms of log-concave vectors

A random vector $X$ with values in a finite-dimensional normed space $(F,\|\ \|)$ is log-concave if its distribution is a log-concave measure. Let $F^*$ be the dual space, with dual norm $\|\varphi\|_*$. For $p\geq 1$, define the weak $p$-th moment of $\|X\|$ by

$$\sup_{\varphi\in F^*,\|\varphi\|_*\leq 1}\left(\mathbb{E}|\varphi(X)|^p\right)^{1/p}.$$

Latała's moment conjecture. There exists a universal constant $C$ such that, for every log-concave vector $X$ with values in a finite-dimensional normed space $(F,\|\ \|)$ and every $p\geq 1$,

$$\left(\mathbb{E}\|X\|^p\right)^{1/p}\leq C\left(\mathbb{E}\|X\|+\sup_{\varphi\in F^*,\|\varphi\|_*\leq 1}\left(\mathbb{E}|\varphi(X)|^p\right)^{1/p}\right).$$

This conjecture seeks to extend Paouris's inequality from the Euclidean norm to arbitrary norms. The source says it was formulated and discussed in Latała's work; the supplied material gives no resolution status.