Functional symmetric $L^p$-Mahler conjecture

Let $p\in (0,\infty]$. For an even function $\phi\in\operatorname{Cvx}(\mathbb{R}^n)$, let $\mathcal{M}_p(\phi)$ denote its $L^p$-Mahler integral.

Functional symmetric $L^p$-Mahler conjecture.

$$\mathcal{M}_p(\phi)\geq \mathcal{M}_p([-1,1]^n).$$

This proposes the functional extension of the symmetric geometric $L^p$-Mahler conjecture, but the source excerpt does not provide a resolution.