Approximate factorization from a small two-terminal connection probability

Let $a$, $b$, and $c$ be vertices, and let $\mathbf{P}(ab\\|c)$, $\mathbf{P}(abc)$, $\mathbf{P}(ac)$, and $\mathbf{P}(bc)$ denote the corresponding bond-percolation event probabilities. Second conjecture. For any $\varepsilon > 0$ there exists $\delta > 0$ such that if $\mathbf{P}(ab\\|c) < \delta$, then

$$\mathbf{P}(abc) - \mathbf{P}(ac)\mathbf{P}(bc) < \varepsilon.$$

This is the stronger of the two proposed stability statements, refining the exact factorization proved when $\mathbf{P}(ab\\|c)=0$; its status is left open in the source.