The simultaneous-crossing independence conjecture for the 4-state Potts model

Take a polygon $\Omega$ whose boundary is composed of edges of a hexagonal lattice of mesh $h$, and let $A_1,A_2,A_3,A_4,A_5,A_6$ be boundary vertices. Assign each hexagon one of four colors independently with probability $1/4$. Define $E_1$ as the event that $A_1A_2$ and $A_4A_5$ are joined by a path of hexagons colored $1$ or $4$, $E_2$ as the analogous event for $A_2A_3$ and $A_5A_6$ using colors $2$ or $4$, and $E_3$ as the analogous event for $A_3A_4$ and $A_6A_1$ using colors $3$ or $4$. Simultaneous-crossing independence conjecture. As $h\to0$ and $(\Omega,A_1,A_2,A_3,A_4,A_5,A_6)$ approaches a planar domain with rectifiable Jordan boundary and six distinct boundary points in the Carathéodory sense, the events become mutually independent; equivalently,

$$P(E_1\cap E_2\cap E_3)-P(E_1)P(E_2)P(E_3)\to0.$$

The conjecture proposes a conformally stable independence phenomenon for the simplest 4-state Potts-model event that does not reduce to two-state events. No proof or disproof is given in the supplied text.