Positive correlation of three-fluid percolation between opposite sides

Let $M_n$ be the finite honeycomb-lattice region with four-colored cells, and let $B_{k,n}\subset\Omega$ be the event that fluid $k$, for $k=1,2,3$, percolates between opposite sides $k$ and $k+3$ of the hexagon $P$. Positive-correlation conjecture. For each $n$,

$$P(B_{1,n}\cap B_{2,n}\cap B_{3,n})\geq P(B_{1,n})P(B_{2,n})P(B_{3,n}).$$

The events are pairwise independent, while the cited theorem shows that their joint dependence vanishes asymptotically. The conjecture asserts positive three-way correlation at every finite scale.