Functional $L^p$-Mahler lower-bound conjecture

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

Functional $L^p$-Mahler lower-bound conjecture.

$$\mathcal{M}_p(\phi)\geq \left(e^{1+\frac1p}p^{\frac1p}\Gamma\left(1+\frac1p\right)\right)^n.$$

This is a universal lower-bound formulation for the functional $L^p$-Mahler integral; the supplied text does not state whether it is solved.