Uniform finite-size criterion for percolation on 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 write $\mathbb P_p^G(o\leftrightarrow S_r)$ for the probability that $o$ is connected to $S_r$ by an open path. Uniform finite-size criterion. For each $d\in\mathbb N$ and $\varepsilon,\delta>0$, there exists $r\in\mathbb N$ such that, for every $G\in\mathcal G_d^*$ and $p\in[0,1]$,

$$\mathbb P_p^G(o\leftrightarrow S_r)\geq\delta\quad\Longrightarrow\quad\mathbb P_{p+\varepsilon}^G(o\leftrightarrow\infty)>0.$$

This is presented as a reformulation of the locality conjecture and is tied to the conjectured continuity of the percolation phase transition. The source says the corresponding continuity question remains open, for example on the three-dimensional cubic lattice.