The automorphism-invariant positive-association percolation criterion

Let $d$ be the dimension, let $\mathbb{Z}^d$ carry a percolation measure $\mathbb{P}$, let $\Lambda_{2N,N,N}$ denote the rectangular box appearing in the crossing event $H(\Lambda_{2N,N,N})$, and write $0 \leftrightarrow \infty$ for the event that the origin belongs to an infinite open cluster. A percolation measure is automorphism invariant if it is invariant under automorphisms of $\mathbb{Z}^d$, and positively associated if increasing events are positively correlated.

Percolation criterion. There are an $\epsilon>0$ and an $N \in \mathbb{N}$ such that, whenever $\mathbb{P}$ is an automorphism-invariant positively associated percolation measure on $\mathbb{Z}^d$ satisfying

$$\mathbb{P}\left(H\left(\Lambda_{2N,N,N}\right)\right)\geq 1-\epsilon,$$

then

$$\mathbb{P}\left(0\leftrightarrow\infty\right)>0.$$

The conjecture is proposed as a sufficient criterion for proving continuity below the critical parameter in the enhancement-percolation setting; the supplied text gives no resolution, so its status remains open.