Critical Riesz energy asymptotic conjecture on spheres

Let $d\geq1$, let $\mathbb{S}^d$ be the unit sphere, and let $\mathcal{H}_d(\mathbb{B}^d)$ and $\mathcal{H}_d(\mathbb{S}^d)$ denote the relevant measures of the unit ball and sphere. Let $C_{s,d}$ be the Riesz-energy coefficient for $s\ne d$. Critical-energy conjecture. As $N\to\infty$,

$$\mathcal{E}_d(\mathbb{S}^d;N)=\frac{\mathcal{H}_d(\mathbb{B}^d)}{\mathcal{H}_d(\mathbb{S}^d)}N^2\log N+C_{d,d}N^2+\mathcal{O}(1),$$

where

$$C_{d,d}=\lim_{s\to d}\left[V_s(\mathbb{S}^d)+\frac{C_{s,d}}{[\mathcal{H}_d(\mathbb{S}^d)]^{s/d}}\right].$$

For $d=2$, the source gives an explicit negative value for $C_{2,2}$. This is the singular boundary case obtained by taking the limit $s\to d$ in the conjectured Riesz asymptotics; its validity remains open.