Convergence to a point or interval conjecture for graph-based generalized Pólya urns

Let $G$ be a finite connected graph, and let $x(n)$ denote the associated urn process. Convergence to a point or interval conjecture. There exists either a point $v(G)$ such that $x(n)$ almost surely converges to $v(G)$, or an interval $(G)$ such that $x(n)$ almost surely converges to a point in $(G)$.

For balanced-bipartite graphs, the corresponding vector field either has no interior equilibrium or has an interval of interior equilibria, while the preceding results give finiteness of the boundary equilibria. The authors cannot prove convergence when an interval of interior equilibria occurs and conjecture that the stated point-or-interval convergence nevertheless holds.