The antipodal-point logarithmic-energy asymptotic conjecture on the sphere

Let $\omega_N\subset\mathbb{S}^2$ be a collection of $N$ antipodal points, meaning that $p\in\omega_N$ implies $-p\in\omega_N$. Let $\mathscr{E}_{\log}(\omega_N)$ denote its logarithmic energy, let $W_{\log}(\mathbb{S}^2)$ denote the continuous logarithmic energy of the sphere, and let $C_{\log}$ be the constant in the unconstrained spherical asymptotic expansion. Antipodal-point asymptotic conjecture. The minimum logarithmic energy among antipodal collections satisfies

$$\min_{\omega_N\subset\mathbb{S}^2}\mathscr{E}_{\log}(\omega_N)=W_{\log}(\mathbb{S}^2)N^2-\frac{1}{2}N\log N+C_{\log}N+o(N).$$

The question is motivated by the correspondence between points of $\mathbb{RP}^2$ and antipodal points on $\mathbb{S}^2$, together with the close upper and lower bounds obtained in the paper. Whether the antipodal constraint preserves the same linear-order constant as the unrestricted spherical problem remains open.