Euclidean-ball asymptotic $L^p$-Mahler conjecture

For $p\in(0,\infty)$, let $B_2^n$ be the Euclidean unit ball and let $\mathcal{M}_p$ denote the $L^p$-Mahler integral or volume as appropriate. The notation $\mathcal{M}_p(|x|^2/2)$ refers to the corresponding quadratic functional.

Euclidean-ball asymptotic conjecture.

$$\mathcal{M}_p(B_2^n)=\mathcal{M}_p(|x|^2/2)e^{o(n)}=\left(2\pi\sqrt{\frac{(1+p)^{\frac{1+p}{p}}}{p}}\right)^n e^{o(n)}.$$

The conjecture would identify the exponential-order asymptotics needed to transfer the geometric Santaló bound to the functional setting; the source notes that the exact formula involves a difficult Bessel-function integral.