Jonckheere–Lou–Bonahon–Baryshnikov vertex-transitivity conjecture

Let $G$ be a large but finite graph with positive curvature. A graph is vertex-transitive when for every pair of vertices there is a graph isomorphism taking one to the other. Let demand and inertia be the vertex quantities defined in the paper. Jonckheere–Lou–Bonahon–Baryshnikov's vertex-transitivity conjecture. If the graph has a vertex-transitive symmetry group, then both the demand and the inertia are constant for all vertices. The paper proves a stronger curvature-independent statement, provided graph isomorphisms also preserve the additional metric $d$; thus the conjecture is resolved under the paper's metric-preservation qualification.