Local density conjecture for domino tilings

Let $p_a$, $p_b$, $p_c$, and $p_d$ be the edge-density quantities associated with the dimer model on an $n\times n$ torus in the thermodynamic limit. Consider an infinite sequence of enlarging planar regions whose normalized boundary height functions converge to a fixed boundary asymptotic height function, and mesoscopic subregions staying away from the boundary. At points where the entropy-maximizing height function has defined partial derivatives, let

$$(s,t)=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right)$$

be the tilt, with $|s|+|t|<2$. Local density conjecture. In the thermodynamic limit, the probability of seeing a domino in a particular location is given by the suitable member of the $4$-tuple $(p_a,p_b,p_c,p_d)$. Equivalently, the local densities of $a$-, $b$-, $c$-, and $d$-edges are $p_a$, $p_b$, $p_c$, and $p_d$, respectively. This conjecture proposes that the torus quantities $p_a,p_b,p_c,p_d$ describe local domino probabilities in planar regions wherever the limiting height function is smooth and the tilt is non-extremal. The source provides no resolution, so the conjecture remains open.