Logarithmic and Coulomb point-set optimality conjecture

Let $E_s(\omega_N)$ denote the Riesz $s$-energy of a configuration $\omega_N\subset\mathbb{S}^2$, let $E_{\log}(\omega_N)$ denote its logarithmic energy, and let $\omega_N^{\log}$ and $\omega_N^{\mathrm{Coul}}$ be the minimal logarithmic-energy and Coulomb point configurations, respectively. Let $\sigma$ be normalized surface area measure, let $\Lambda_2$ be the regular triangular lattice, and let $\zeta_{\Lambda_2}$ be its Epstein–Zeta function. Logarithmic and Coulomb optimality conjecture. For all $-2<s<4$, $s\neq 0$, the minimal logarithmic-energy points and Coulomb points are asymptotically minimal to second-term precision: with $\omega_N=\omega_N^{\log}$ or $\omega_N=\omega_N^{\mathrm{Coul}}$, either

$$\lim_{N\to\infty}\frac{E_s(\omega_N)-\mathcal{I}_s[\sigma]N^2}{\mathcal{E}_s(N)-\mathcal{I}_s[\sigma]N^2}=1$$

or

$$\lim_{N\to\infty}\frac{E_s(\omega_N)-\frac{(\sqrt{3}/2)^{s/2}\zeta_{\Lambda_2}(s)}{(4\pi)^{s/2}}N^{1+s/2}}{\mathcal{E}_s(N)-\frac{(\sqrt{3}/2)^{s/2}\zeta_{\Lambda_2}(s)}{(4\pi)^{s/2}}N^{1+s/2}}=1.$$

Furthermore, Coulomb points are minimal for logarithmic energy to third-term precision:

$$\lim_{N\to\infty}\frac{E_{\log}(\omega_N^{\mathrm{Coul}})-(1/2-\log2)N^2+(1/2)N\log N}{\mathcal{E}_{\log}(N)-(1/2-\log2)N^2+(1/2)N\log N}=1.$$

The claim is explicitly conditional on the preceding Riesz-energy conjectures for the relevant ranges; the source reports numerical support but no proof.