Conjecture on the unique extremal vertex in negatively curved symmetric graphs

Let $G$ be a large but finite graph with negative curvature. Suppose its symmetric group fixes a unique point $v_0$. Let the inertia and demand of a vertex be the quantities defined in the paper. Unique-extremal-vertex conjecture. The point $v_0$ is the unique point of minimum inertia and maximum demand. The supplied context does not provide a resolution or a proof of this claim.