The asymptotic spherical-distribution conjecture for the angular sum

Let $\mathcal{Q}_N$ denote the configuration space of $N$ points in three-dimensional space, and let $F_N$ be the angular sum defined in the paper. The asymptotic spherical-distribution conjecture. As $N\to\infty$, the configuration maximizing $F_N$ consists of $N$ points uniformly distributed on a sphere, and

$$\sup_{\mathcal{Q}_N}F_N\sim\frac{2}{9}N^3+o(N^3).$$

This conjecture describes the large-$N$ behavior of the geometric optimization problem and proposes both the limiting distribution and the leading asymptotic growth of the maximum.