Uniqueness and stability conjecture for spherical graph representations

A stable representation is an ${\mathcal{M}}$-representation that is a local minimum with respect to the ordering relation defining stability. Consider a graph on the sphere and fix the set of edges crossing a given equator. Spherical representation conjecture. Each such graph has at most one non-trivial stable representation up to symmetry transformations on the sphere. Furthermore, jammed configurations are stable. This conjecture would restrict the structure of stable representations on the sphere and connect them directly with jammed configurations used in constructing maximal disk packings. The source does not state whether it has been resolved.