Next-order Riesz energy asymptotic conjecture for spheres

Let $d\geq 2$, let $\mathbb{S}^d$ be the unit sphere, let $\mathcal{H}_d(\mathbb{S}^d)$ denote its $d$-dimensional Hausdorff measure, and let $V_s(\mathbb{S}^d)$ denote its continuous $s$-energy. Let $C_{s,d}$ be the hypersingular Riesz-energy constant for $s>d$. Next-order Riesz energy conjecture. For $-2<s<d+2$, $s\ne d$, there is a constant $C_{s,d}$ such that

$$\mathcal{E}_s(\mathbb{S}^d;N)=V_s(\mathbb{S}^d)N^2+\frac{C_{s,d}}{[\mathcal{H}_d(\mathbb{S}^d)]^{s/d}}N^{1+s/d}+o(N^{1+s/d})\quad\text{as }N\to\infty,$$

where, for $s>d$, $C_{s,d}$ is the same constant as in the hypersingular energy problem. Furthermore, for $d=2,4,8,24$, $C_{s,d}=|\Lambda_d|^{s/d}\zeta_{\Lambda_d}(s)$, with $\Lambda_d$ respectively the hexagonal lattice, $D_4$, $E_8$, and the Leech lattice. This conjecture proposes the two leading terms across the potential-theoretic and hypersingular regimes, but the source notes that higher-order terms may fail to exist in the proposed form.