Hypersingular Riesz energy asymptotic conjecture for exponents between two and four

For $s\in(2,4)$, let ${\mathpzc E}_s(\mathbb{S}^2,N)$ be the minimal total Riesz $s$-energy, let $W_s$ be the analytically continued continuum coefficient, and let $\operatorname{\zeta}_\Lambda(s)$ be the hexagonal-lattice zeta function. Hypersingular asymptotic conjecture. As $N\to\infty$,

$${\mathpzc E}_s(\mathbb{S}^2,N)\sim\left(\frac{\sqrt3}{8\pi}\right)^{s/2}\operatorname{\zeta}_\Lambda(s)N^{1+s/2}+W_sN^2+o(N^2).$$

The first term is expected from the hexagonal-lattice constant, while $W_sN^2$ is the next-to-leading term; the statement is listed among open problems.