The torus-to-plane convergence conjecture for domino configurations

Let $(s,t)$ satisfy $|s|+|t| \leq 2$, and let $a,b,c,d$ be weights satisfying $ab=cd$ such that the average height function for weighted torus tilings has tilt $(s,t)$. Let $\mu_{s,t}$ be the associated measure on domino tilings of the plane, invariant under color-preserving translations and conditionally uniform. Torus-to-plane convergence conjecture. For any colored configuration of dominos, the probability of finding it in a specified location in a random $n \times n$ torus tiling converges as $n \rightarrow \infty$ to the value given by the measure $\mu_{s,t}$. Proposition coupling-function calculations approach this claim but, as stated in the paper, restrict $n$ to a large subset of the integers rather than all sufficiently large $n$.