Fundamental asymptotic conjecture for subcritical Riesz energy

For $-2<s<2$ with $s\neq0$, let $u_s(N)$ be the minimal average unstandardized Riesz pair-energy on $\mathbb{S}^2$, let $W_s$ denote the continuum energy, and let $C_s$ be a constant. Subcritical Riesz asymptotic conjecture. There exists a function $\Omega_s(N)$ such that

$$u_s(N)=\frac{1}{s}\frac{C_s}{(4\pi)^{s/2}}N^{s/2-1}+\frac{W_s}{s}N^{-1}+\Omega_s(N),\qquad N\geq2,$$

and $N^{1-s/2}\Omega_s(N)\to0$ as $N\to\infty$. Equivalently, the associated total energy has leading terms $W_sN^2+C_s(4\pi)^{-s/2}N^{1+s/2}$ with a lower-order remainder. This is described as a fundamental conjecture; the rate and possible oscillation of the remainder remain unclear.