The variational-principle conjecture for finite domino configurations

Consider a sequence of finite regions converging to a fixed shape, and let $(s,t)$ be the tilt given by the variational principle wherever that tilt is defined. Let $\mu_{s,t}$ be the associated measure on plane domino tilings. Finite-region configuration conjecture. In the thermodynamic limit for a sequence of finite regions converging to a fixed shape, the probability of seeing any colored configuration of dominos is given by the measure $\mu_{s,t}$, wherever the tilt $(s,t)$ given by the variational principle is defined and satisfies $|s|+|t|<2$. This is presented as a further claim conditional on the torus-to-plane convergence conjecture and is described in the source as an open problem connecting the variational principle to local configuration probabilities.