The $L^2$ boundedness conjecture for nonamenable quasi-transitive graphs

Let $G$ be a connected, locally finite, nonamenable, quasi-transitive graph. Let $T_p$ be the two-point matrix of Bernoulli bond percolation, with entries $T_p(u,v)=\mathbb P_p(u\leftrightarrow v)$, and define

$$p_{2\to 2}(G)=\sup\{p\in[0,1]:\|T_p\|_{2\to 2}<\infty\}.$$

$L^2$ boundedness conjecture.

$$p_c(G)<p_{2\to 2}(G).$$

This strong quantitative conjecture implies both the separation of the critical and uniqueness thresholds and the triangle condition. It is known in a variety of special cases but remains open in general.