The four-point inequality conjecture for Atiyah–Sutcliffe angular quantities

For four distinct points $x_1,x_2,x_3,x_4$ in $\mathbb{R}^3$, let the inequality denoted by

$$in the source be the angular inequality established for the relevant tetrahedral configuration. **Four-point angular inequality conjecture.** The inequality$$

holds for every such four-point configuration.

The paper notes that this inequality extends beyond the convex-quadrilateral case and reports verification for several million random tetrahedra, but leaves the general assertion conjectural.