Rakhmanov–Saff–Zhou logarithmic energy asymptotic conjecture for the sphere

Let $\mathcal{E}_{\mathrm{log}}(\mathbb{S}^2;N)$ denote the optimal logarithmic energy of $N$ points on the unit sphere, and let $C_{\mathrm{log},2}$ and $D_{\mathrm{log},2}$ be constants independent of $N$. Rakhmanov–Saff–Zhou's conjecture. There exist constants $C_{\mathrm{log},2}$ and $D_{\mathrm{log},2}$ such that

$$\mathcal{E}_{\mathrm{log}}(\mathbb{S}^2; N) = \left( \frac{1}{2} - \log 2 \right) N^2 - \frac{1}{2} N \log N + C_{\mathrm{log},2} \, N + D_{\mathrm{log},2} \, \log N + \mathcal{O}(1) \quad \text{as $N \to \infty$.}$$

This refines the known upper and lower bounds for the linear-order remainder in the logarithmic energy asymptotics.