Gaussian energy Mellin-type identity conjecture in dimensions 8 and 24

Let $\varphi(x)=e^{-c|x|^2}$ in $\mathbb R^n$, where $n\in\{8,24\}$, and let $h$ be the limiting auxiliary function from the sharp energy-bound convergence conjecture. Let $\Lambda_n$ denote $E_8$ for $n=8$ and the Leech lattice for $n=24$, and set $\psi(x)=|x|^2\varphi(x)$. Gaussian energy identity conjecture.

$$\widehat h(0)=\frac{2c}{n}E_\psi(\Lambda_n).$$

This predicts an exact relation between the limiting linear-programming bound and the energy of the exceptional lattice for Gaussian potentials; the source provides it as a conjectural analogue of the earlier limiting identities.