Universal sharpness of the four-point bound for five particles on the sphere

Let $E_2$ denote the second relaxation, or four-point bound, in the semidefinite hierarchy for energy minimization. For five particles on $S^2$, consider all completely monotonic pair potentials in the squared chordal distance. Universal sharpness conjecture. The bound $E_2$ is universally sharp for five particles on $S^2$, meaning that it gives the ground-state energy for every such pair potential. This would imply sharpness across the phase transition between the triangular-bipyramid and square-pyramid minimizers, but the conjecture is not proved.