Positive correlation of three fluids percolating from the center

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 $M_n$, with the color of $O$ disregarded. Center-percolation correlation conjecture. For each $n$,

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

This is the center analogue of the conjecture for percolation between opposite sides. The paper presents it as an open conjecture and discusses a stronger asymptotic ratio conjecture separately.