Uniqueness conjecture for stressed geodesic graphs on the sphere

Let $G$ be a vertex $3$-connected planar graph embedded in the unit-radius sphere. Assume that each edge is a geodesic of length less than $pi$ and each face has total perimeter less than $2pi$, and that there is an equilibrium stress positive on every edge. Consider degree-one maps of the sphere to itself extending the map from the original realization, with each geodesic edge constrained not to increase in length. Uniqueness conjecture for stressed geodesic graphs. Under these conditions, the configuration is unique up to rotation. The conjecture asks whether every such stressed spherical graph has a unique realization in the specified degree-one class; the source reports no counterexample, but does not establish the claim in general.