The post-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 post-FKG conjecture.

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

This is an FKG-like inequality proposed as an approach to proving $\theta(p_c)=0$ in percolation. The authors state that they could neither prove nor disprove it, with only restricted numerical evidence and proofs in several special cases.