Single-survivor conjecture for competing urns on hypercubes and discrete tori

Let $K\ge3$, let the underlying graph be either a hypercube or a discrete torus, and let the initial configuration be any nonzero configuration of the $K$-type competing urn scheme. Single-survivor conjecture. The process has almost surely a single surviving type. The path and cycle are the only finite connected graphs known in the source to have this property for every $K\ge3$, while coexistence is known on some other graphs; the conjecture remains open for hypercubes and discrete tori.