Uniqueness conjecture for locally minimal spherical graph configurations

For each $3$-connected embedded graph $(G,\mathbf p)$ on the unit sphere with topological degree one and with every facial perimeter of length less than $2\pi$, define a partial order by declaring $(G,\mathbf p)\leq(G,\mathbf q)$ when every edge has geodesic length in $\mathbf p$ at most its geodesic length in $\mathbf q$. A configuration $(G,\mathbf p)$ is locally minimal if it is locally minimal with respect to this partial order. Locally minimal uniqueness conjecture. Any locally minimal $(G,\mathbf p)$ is unique in its homotopy class up to rotation with respect to the partial order. This is a proposed semi-global uniqueness principle for degree-one geodesic configurations on the sphere; the source does not give a proof or a counterexample.