Jonckheere–Lou–Bonahon–Baryshnikov conjecture on curvature and balance

Let $G$ be a large but finite graph with positive curvature. Let the demand and inertia of a vertex be the graph quantities defined in the paper. Jonckheere–Lou–Bonahon–Baryshnikov's balance conjecture. The graph has more balanced values for demand and inertia than a graph with negative curvature. The surrounding text gives positive-curvature counterexamples with highly unbalanced demand and inertia, so the formulation is not supported by those examples.