Jonckheere–Lou–Bonahon–Baryshnikov conjecture on highest demand vertices

Let $G$ be a large but finite graph with negative curvature. Let the demand of a vertex mean the quantity measuring how many shortest paths pass through it, as defined in the paper. Jonckheere–Lou–Bonahon–Baryshnikov's highest-demand conjecture. There are very few vertices with the highest demand. The paper discusses counterexamples and limitations of this formulation, including graphs with uniformly distributed or infinite demand.