The strict inequality conjecture for critical and unification probabilities on non-amenable graphs

Let $\Gamma$ be a connected, locally finite graph. The critical probability is

$$p_\mathrm c(\Gamma):=\inf\{p\in[0,1]:\text{with positive probability }\omega\text{ has some infinite cluster}\},$$

and the unification probability is

$$p_\mathrm u(\Gamma):=\inf\{p\in[0,1]:\text{almost surely there is a unique infinite cluster in }\omega\}.$$

A graph is non-amenable if there is a constant $\Phi>0$ such that every non-empty finite vertex set $K$ satisfies $|\operatorname{bd} K|\geq \Phi|K|$, and it is quasi-transitive if its automorphism group has finitely many vertex orbits.

Strict inequality conjecture. If $\Gamma$ is non-amenable and quasi-transitive, then

$$p_\mathrm c(\Gamma)<p_\mathrm u(\Gamma).$$

The inequality asserts that, on every non-amenable quasi-transitive graph, there is a nonempty interval of parameters in which infinitely many infinite clusters coexist. The converse to the known fact that amenable transitive graphs have at most one infinite cluster almost surely would establish a sharp distinction between amenable and non-amenable percolation, but the conjecture remains open in the source.