Strict asymptotic enhancement of three-fluid center percolation

Let $M_n$ be the finite honeycomb-lattice region, let $O$ be its central cell, and let $A_{k,n}\subset\Omega$ be the event that fluid $k$, for $k=1,2,3$, percolates from $O$ to the boundary of $M_n$. Asymptotic enhancement conjecture.

$$\lim_{n\to+\infty}\frac{P(A_{1,n}\cap A_{2,n}\cap A_{3,n})}{P(A_{1,n})P(A_{2,n})P(A_{3,n})}>1.$$

This conjecture is stronger than finite-scale positive correlation: it predicts a strictly nontrivial limiting enhancement of the joint probability. The paper gives informal arguments in its favor but does not prove it.