Logarithmic energy asymptotic conjecture on the sphere

Let $u_0(N)$ be the minimal average logarithmic pair-energy of $N$ points on $\mathbb{S}^2$, let $W_{\mathrm{log}}$ be its continuum contribution, and let $C_{\mathrm{log}}$ be the stated constant. Logarithmic asymptotic conjecture. There exists a function $\Omega_0(N)$ such that

$$u_0(N)=-\frac{\log N}{2N}+\frac{W_{\mathrm{log}}+C_{\mathrm{log}}}{N}-\frac{\log N}{2N^2}+\frac{W_{\mathrm{log}}+C_{\mathrm{log}}}{N^2}+\Omega_0(N),\qquad N\geq2,$$

with $N\Omega_0(N)\to0$ as $N\to\infty$, or in the stronger proposed form $N^2\Omega_0(N)\to c_0\neq0$, where

$$C_{\mathrm{log}}=2\log2+\frac12\log\frac23+3\log\frac{\sqrt\pi}{\Gamma(1/3)}=-0.05560530494339251850\ldots<0.$$

The conjecture is the logarithmic specialization of the large-$N$ energy expansion; the stronger remainder statement is explicitly presented as an alternative form.