Lattice optimality conjecture for Riesz energy constants in dimensions 2, 4, 8, and 24

Let $C_{s,d}$ be the asymptotic Riesz $s$-energy constant for $d$-rectifiable sets, and let

$$\zeta_\Lambda(s)=\sum_{0\neq x\in\Lambda}|x|^{-s}$$

be the Epstein zeta function of a lattice $\Lambda\subset\mathbb{R}^d$ with covolume $|\Lambda|>0$. Define

$$\widetilde{C}_{s,d}:=|\Lambda_d|^{s/d}\zeta_{\Lambda_d}(s),$$

where $\Lambda_2$ is the equi-triangular lattice, $\Lambda_4$ is the $D_4$ lattice, $\Lambda_8$ is the $E_8$ lattice, and $\Lambda_{24}$ is the Leech lattice. Lattice optimality conjecture. For $d=2,4,8,$ and $24$, and $s>d$,

$$C_{s,d}=\widetilde{C}_{s,d}=|\Lambda_d|^{s/d}\zeta_{\Lambda_d}(s).$$

The claim concerns equality in the general lattice upper bound for $C_{s,d}$ and asserts that the indicated highly symmetric lattices attain the asymptotic energy constant. Its status is unresolved in the supplied source.