A conjecture implying $\theta (p_c)=0$ for Bernoulli bond percolation on ${\mathbb{Z}}^d$

We introduce a conjecture for Bernoulli bond percolation on finite graphs. Roughly speaking, it asserts that if each boundary vertex is associated with a highly probable event that is increasing with respect to the percolation configuration, then, conditionally on the origin being connected to the boundary, the origin is likely to be connected to a boundary vertex whose associated event occurs. The conjecture implies $\theta (p_c)=0$ for Bernoulli bond percolation on ${\mathbb{Z}}^d$, for every $d\ge 2$. We prove the conjecture for finite planar graphs when the origin and the boundary vertices lie on the outer-face boundary, with the explicit bound $1-2\sqrt{\epsilon }$. The proof combines a left-first depth-first exploration with a partial FKG inequality adapted to monotonicity up to a stopping time. A counterexample shows that the connectivity structure is essential: the analogous statement fails when the connectivity events are replaced by arbitrary increasing events.