The uniqueness conjecture for positive-entropy domino-tiling measures

Let $\mu_{s,t}$ denote the measure on domino tilings of the plane associated with tilt $(s,t)$. A measure is conditionally uniform when, given a tiling outside any finite region, the conditional distribution inside that region is uniform; require also invariance under color-preserving translations, ergodicity, and positive entropy. Uniqueness conjecture. Every ergodic, conditionally uniform measure on the set of tilings of the plane that is invariant under color-preserving translations and has positive entropy is of the form $\mu_{s,t}$ for some $(s,t)$ satisfying $|s|+|t|<2$. The paper notes that the measures $\mu_{s,t}$ are mixing and hence ergodic in the positive-entropy regime, but says that their characterization by these properties cannot be proved.