Bessel-integral asymptotic conjecture for the Euclidean ball

For $p\in(0,\infty)$ and $I_{n/2}$ the modified Bessel function appearing in the explicit formula for $\mathcal{M}_p(B_2^n)$,

Bessel-integral asymptotic conjecture.

$$\int_0^\infty \frac{r^{n+\frac{n}{2p}-1}}{I_{n/2}(r)^{\frac1p}}\,dr=\left(\frac{n^{1+\frac{1}{2p}}}{e^{1+\frac{1}{2p}}}\sqrt{p(1+p)^{1+\frac{1}{p}}}\right)^n e^{o(n)}.$$

The source states this as the equivalent integral formulation of the Euclidean-ball asymptotic conjecture, reducing the problem to estimating a Bessel-function integral.