Strict $q$-to-$q$ threshold conjecture for nonamenable quasi-transitive graphs

Let $G$ be a connected, locally finite, quasi-transitive, nonamenable graph. For $q\in(1,\infty)$, let $p_{q\to q}(G)$ denote the $q$-to-$q$ threshold used in the paper.

Strict $q$-to-$q$ threshold conjecture. For every $q\in(1,\infty)$,

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

This is presented as a strengthening of the Benjamini–Schramm and triangle-condition conjectures. The paper notes that it is equivalent to $p_c<p_{2\to2}$ under the same hypotheses, while the general assertion remains open.