The asymptotic expansion conjecture for minimal Riesz 2-energy on the sphere

Let $\mathbb{S}^2$ be the unit sphere and let $\mathcal{E}_2(\mathbb{S}^2;N)$ denote the minimal Riesz $2$-energy of $N$ points on $\mathbb{S}^2$. Let $\gamma_n(\alpha)$ be the generalized Stieltjes constant appearing as the coefficient $\frac{\gamma_n(\alpha)}{n!}$ of $(1-s)^n$ in the Laurent expansion of the Hurwitz zeta function $\zeta(s,\alpha)$ about $s=1$. The asymptotic expansion conjecture.

$$\mathcal{E}_2(\mathbb{S}^2;N)=\frac{1}{4}N^2\log N+C_{2,2}N^2+o(N^2),$$

where

$$C_{2,2}=\frac{1}{4}\left(\gamma-\log(2\sqrt{3}\pi)\right)+\frac{\sqrt{3}}{4\pi}\left(\gamma_1(\tfrac{2}{3})-\gamma_1(\tfrac{1}{3})\right).$$

This conjectural asymptotic is used to sharpen lower bounds for logarithmic energy on the sphere. The source does not provide evidence of a resolution.