Conjecture on comparable probabilities of essential and influential edges

For an edge $j$, let $E_j$ be the event that every geodesic passes through $j$, let $\hat E_j$ be the event that at least one geodesic passes through $j$, let $A_j$ be the event that $\partial_j f(\omega)\neq 0$, and let $\hat A_j$ be the event that $\partial_j f(\omega)=b-a$. The paper considers these events in the first-passage percolation environment.

Comparability conjecture. There exists a constant $C$ independent of $n$ such that

$$\mathbb P(A_j)\leq C\mathbb P(\hat A_j),\qquad \mathbb P(\hat E_j)\leq C\mathbb P(E_j),\qquad \mathbb P(\hat E_j)\leq C\mathbb P(\hat A_j).$$

The conjecture would show that the probabilities of the four edge categories are comparable in the stated directions. The paper notes that the result is natural but was not needed there and that its proof is not obvious.