The eight-point maximum conjecture for the three-dimensional angular sum

Let $\mathcal{Q}_N$ denote the configuration space of $N$ points in $d$-dimensional space, and let $F_N$ be the angular sum defined in the paper. The eight-point maximum conjecture. For $d=3$,

$$\sup_{\mathcal{Q}_8}F_8=16\left[\frac{4}{5}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{5}}+\frac{4(1+\sqrt{2})}{\sqrt{5}\sqrt{5+4\sqrt{2}}}+\frac{3+2\sqrt{2}}{\sqrt{5+4\sqrt{2}}}-\frac{1}{5+4\sqrt{2}}\right]\simeq79.501.$$

The maximum is attained when the eight points form two identical unit-edge squares in parallel planes separated by $h=\sqrt{1+2\sqrt{2}}/2$, with the axis joining their centers perpendicular to the squares and the squares rotated relative to one another by $\pi/4$. This is a numerical conjecture about the maximizing configuration; the paper compares it with the cube, whose value is slightly smaller.