Uniform subcritical decay criterion for non-one-dimensional transitive graphs

Let $\mathcal G_d^*$ be the class of infinite transitive graphs of degree $d$ that are not one-dimensional. For a graph $G$ with origin $o$, let $S_r$ be the sphere of radius $r$ around $o$, and let $p_c(G)$ be its Bernoulli bond percolation critical probability. Uniform subcritical decay conjecture. For each $d\in\mathbb N$ and $\varepsilon>0$, there exists a function $h_\varepsilon=h_{d,\varepsilon}:\mathbb N\to(0,\infty)$ with $h_\varepsilon(r)\to0$ as $r\to\infty$ such that, for every $G\in\mathcal G_d^*$ and $r\geq1$,

$$\mathbb P_{p_c(G)-\varepsilon}^G(o\leftrightarrow S_r)\leq h_\varepsilon(r).$$

This is another reformulation of the locality conjecture, expressing uniform decay away from criticality. The source explicitly notes that its proof does not obtain an explicit function $h_\varepsilon$ and that quantitative bounds are deferred to future work.