The tree-coloring strong spatial mixing conjecture

Let $T$) be a tree of maximum degree $\Delta$, and let $q$ be the number of colors. The uniform distribution on proper $q$-colorings of $T$ is considered with strong spatial mixing (SSM), meaning that the influence of differing boundary pinnings on a vertex's marginal decays exponentially in their distance. Tree-coloring SSM conjecture. For $q \geq \Delta+1$, the uniform distribution on $q$-colorings on trees of maximum degree $\Delta$ exhibits strong spatial mixing with exponential decay rate. This is a basic unresolved question even on trees and is closely connected to understanding correlations in random colorings and to algorithmic sampling.