Conjecture on ergodic Gibbs measures and free energy for quadri-tilings

Fix $(s,t,p,q)\in{\mathbb R}^4$. Let ${\cal Q}_n^{(s,t,p,q)}$ be the quadri-tilings of ${\cal Q}_n$ with first height change $(\lfloor ns\rfloor,\lfloor nt\rfloor)$ and second height change $(\lfloor np\rfloor,\lfloor nq\rfloor)$, and let $\mu_n^{(s,t,p,q)}$ be the conditional Boltzmann measure on this set. Let $\mu_n$ be the unconditioned Boltzmann measure, and let $\mu^{(s,t,p,q)}$ denote a Gibbs measure of slope $(s,t,p,q)$. Ergodic Gibbs-measure and free-energy conjecture. For each $(s,t,p,q)$ for which ${\cal Q}_n^{(s,t,p,q)}$ is nonempty for $n$ sufficiently large, $\mu_n^{(s,t,p,q)}$ converges as $n\to\infty$ to an ergodic Gibbs measure $\mu^{(s,t,p,q)}$ of slope $(s,t,p,q)$. Furthermore, $\mu_n$ converges to $\mu^{(s_0,t_0,p_0,q_0)}$, where $(s_0,t_0,p_0,q_0)$ is the limit of the slopes of $\mu_n$. If $(s_0,t_0,p_0,q_0)$ lies in the interior of the set of $(s,t,p,q)$ for which ${\cal Q}_n^{(s,t,p,q)}$ is nonempty for $n$ sufficiently large, then every ergodic Gibbs measure of slope $(s,t,p,q)$ is of the form $\mu^{(s,t,p,q)}$ for some $(s,t,p,q)$ as above; equivalently, $\mu^{(s,t,p,q)}$ is the unique ergodic Gibbs measure of that slope. Moreover, $\mu^{(s_0,t_0,p_0,q_0)}$ is the unique measure with minimal total free energy per fundamental domain. This conjecture concerns the existence and uniqueness of infinite-volume ergodic Gibbs measures at each admissible slope and the selection of the minimal-free-energy measure; the source cites results of Sheffield and Kenyon–Okounkov–Sheffield as motivation but gives no resolution.