The uniform continuity consequence of the pre-FKG conjecture

Let $G$ be a graph, let $A\subset G$, and let $0,b\in G$ be vertices. The uniform continuity conjecture. For every $\varepsilon>0$ there exists $\delta>0$ such that, whenever

$$\mathbb{P}(0\leftrightarrow A)>1-\delta$$

and

$$\mathbb{P}(a\leftrightarrow b)>1-\delta$$

for all $a\in A$, one has

$$\mathbb{P}(0\leftrightarrow b)>1-\varepsilon.$$

This is introduced as a weaker version of the post-FKG conjecture and is sufficient for the paper's proof that $\theta(p_c)=0$ in every dimension. Its resolution is not given.