Small two-terminal connection probabilities force a negligible three-way connection

Let $a$, $b$, and $c$ be vertices, and let $\mathbf{P}(ab\\|c)$, $\mathbf{P}(ac\\|b)$, $\mathbf{P}(abc)$, and $\mathbf{P}(a\\|b\\|c)$ denote the corresponding bond-percolation event probabilities. First conjecture. For any $\varepsilon > 0$ there exists $\delta > 0$ such that if $\mathbf{P}(ab\\|c) < \delta$ and $\mathbf{P}(ac\\|b) < \delta$, then $\mathbf{P}(abc)$ or $\mathbf{P}(a\\|b\\|c)$ is less than $\varepsilon$. This asks for a stability version of the zero-probability implication established earlier: simultaneous smallness of the two connection probabilities should force one of the two remaining probabilities to be small.