Conjecture on the thermodynamic limit of quadri-tiling Boltzmann measures

Let ${\mathbb T}$ be a periodically tiled torus with lattice of periods $\Lambda$, and let ${\mathbb T}_n={\mathbb T}/n\Lambda$ be such that ${\mathbb T}_n^*$ is bipartite. Let ${\cal Q}_n$ be the triangular quadri-tilings whose underlying tiling is a lozenge tiling of ${\mathbb T}_n$, and let $\mu_n$ be their Boltzmann measures. Suppose a critical weight function is assigned to quadri-tiles. Thermodynamic-limit conjecture. The Gibbs measure of Corollary 4 is the limit of the Boltzmann measures $\mu_n$. This predicts that the infinite-volume Gibbs measure constructed from the lozenge and triangular tiling measures is obtained as the thermodynamic limit of the finite-volume Boltzmann measures; the source gives no resolution of the conjecture.