The pre-FKG conjecture for connection probabilities

Let $G$ be a finite graph with arbitrary probabilities on its edges. Let $0,b\in G$ be vertices and let $A\subset G$ be a set of vertices. The pre-FKG conjecture.

$$\mathbb{P}(0\leftrightarrow b)\ge \min\{\mathbb{P}(0\leftrightarrow A, a\leftrightarrow b):a\in A\}.$$

This is a formally stronger inequality than the post-FKG conjecture, from which the latter follows by the FKG inequality. The authors present it as an important possible strengthening and state that it remains unproved.