Asymptotic $L^p$-Mahler volume conjecture for the simplex

For $p\in(0,\infty)$, let $\Delta_{n,0}$ be the centered simplex and let $\mathcal{M}_p(\Delta_{n,0})$ denote its $L^p$-Mahler volume.

Simplex asymptotic conjecture.

$$\mathcal{M}_p(\Delta_{n,0})=\left(e^{1+\frac1p}p^{\frac1p}\Gamma\left(1+\frac1p\right)\right)^n e^{o(n)}.$$

The conjecture predicts the dimensional asymptotics of the geometric simplex and is motivated by the corresponding functional simplex calculation.