Five-point triangular bipyramid or square-pyramid energy conjecture

Let $S^2$ be the unit sphere, and let a completely monotonic potential function be a potential function whose derivatives alternate in sign on the relevant domain. For five points on $S^2$, consider the energy obtained by summing the potential over all pairwise distances. Five-point energy conjecture. For each completely monotonic potential function, either the triangular bipyramid or a square pyramid minimizes energy for $5$ points in $S^2$. This concerns the first number of points for which universal optimality fails; the triangular bipyramid is an equilibrium for every potential, while the optimal square-pyramid latitude may depend on the potential function.