Entropy factorization conjecture for cone measures on ℓ_p-spheres

For $p>0$, let ${\mathbb{S}}^{n-1}_p=\{x\in\mathbb{R}^n:\|x\|_p=1\}$, where $\|x\|_p=(\sum_{i=1}^n|x_i|^p)^{1/p}$. Let $\mathbf{P}$ be the cone measure on ${\mathbb{S}}^{n-1}_p$, and let $\theta$ be a probability vector with associated quantities $\theta_{\star\star}$ and $\theta_A$. For $f\in L\log L$, write $\operatorname{Ent}(f)$ for entropy and $\operatorname{Ent}_A(f)$ for the entropy conditional on the coordinates indexed by $A$. Entropy factorization conjecture. For all $p>0$, all probability vectors $\theta$, and all $f\in L\log L$, the cone measure satisfies

$$\theta_{\star\star}\,\operatorname{Ent}(f)\leq\sum_{A\subset[n]}\theta_A\mathbf{E}[\operatorname{Ent}_A(f)].$$

The claim would extend the entropy-factorization estimates proved for the Euclidean and $\ell_1$ spheres to all positive $p$, and the authors suggest it should follow from their main theorem; its status is not established in the supplied text.