Comparable weak and strong moments for symmetric log-concave measures

Let $\mu$ be a probability measure on $\mathbb R^n$, and let $\\|\cdot\\|$ be any norm on $\mathbb R^n$ with dual norm $\\|\cdot\\|_*$. For $p\geq q\geq 2$, comparable weak and strong moments with constant $\gamma$ means

$$\left(\int \left|\\|x\\|-\left(\int \\|x\\|^q\\,d\mu\right)^{1/q}\right|^p d\mu\right)^{1/p} \leq \gamma\sup_{\\|u\\|_*\leq 1}\left(\int |\langle u,x\rangle|^p d\mu\right)^{1/p}.$$

Comparable weak and strong moments conjecture. Every symmetric log-concave probability measure on $\mathbb R^n$ satisfies $\mathrm{CWSM}(C)$.

The preceding proposition derives a comparable-moments estimate from the concentration inequality, so the conjecture would follow from the paper's broader concentration conjecture. The supplied text does not state a resolution.