Hutchcroft's weighted-amenability threshold conjecture

Let $G$ be a graph, let $\Gamma$ be a closed subgroup of $\operatorname{Aut}(G)$ acting transitively on $G$, and let $\mathbf{w}$ be the induced relative weight function. Let $p_h(G,\Gamma)$ be the heavy-cluster threshold and let $p_u(G)$ be the uniqueness threshold. Hutchcroft's conjecture. The graph $G$ is $\mathbf{w}$-amenable if and only if

$$p_h(G,\Gamma)=p_u(G).$$

This is presented as a generalization of the Benjamini–Schramm conjecture relating amenability to equality of percolation thresholds. Its status is not resolved in the supplied text.