Jonckheere–Lou–Bonahon–Baryshnikov conjecture on demand and inertia

Let $G$ be a large but finite graph with negative curvature. Let the demand and inertia of a vertex be the graph quantities defined in the paper. Jonckheere–Lou–Bonahon–Baryshnikov's demand–inertia conjecture. The vertices with the highest demand are near the vertices with the smallest inertia. The paper presents examples where the two types of vertices can be far apart, so the proposed relationship is challenged by the subsequent discussion.