Triangular-lattice expansion conjecture for hypersingular Riesz energy

Let $\mathcal{E}_s(N)$ be the minimal Riesz $s$-energy on $\mathbb{S}^2$, let $\Lambda_2$ be the regular triangular lattice in $\mathbb{R}^2$, and set

$$V_s=\frac{2^{1-s}}{2-s},\qquad s\in\mathbb{C}\setminus\{2\}.$$

Hypersingular triangular-lattice expansion conjecture. For $2<s<4$,

$$\mathcal{E}_s(N)=\frac{(\sqrt{3}/2)^{s/2}\zeta_{\Lambda_2}(s)}{(4\pi)^{s/2}}N^{1+s/2}+V_sN^2+o(N^2).$$

The known result gives only the leading-order growth $\mathcal{E}_s(N)\asymp N^{1+s/2}$; this conjecture predicts both the triangular-lattice leading coefficient and the analytically continued continuous-energy term.