The triangle condition for critical percolation on nonamenable graphs

Let $G$ be a connected, locally finite, nonamenable, quasi-transitive graph, and let $\mathbb P_p(u\leftrightarrow v)$ denote the probability that $u$ and $v$ are connected in Bernoulli bond percolation at parameter $p$. Define the triangle diagram at the critical probability by

$$\nabla_{p_c}(v)=\sum_{u,w\in V}\mathbb P_{p_c}(v\leftrightarrow u)\mathbb P_{p_c}(u\leftrightarrow w)\mathbb P_{p_c}(w\leftrightarrow v).$$

Triangle-condition conjecture. For every $v\in V$,

$$\nabla_{p_c}(v)<\infty.$$

The triangle condition is a standard sufficient condition for mean-field critical exponents, including the expected exponents for cluster volume and percolation probability. It is known in several special cases but remains completely open for general nonamenable quasi-transitive graphs.