Kenyon–Okounkov conjecture for height fluctuations in random tilings

Let $\mathsf{Ht}_\mathcal N$ be the random height function of the model at scale $\mathcal N$, let $\mathbb E[\mathsf{Ht}_\mathcal N]$ be its expectation, let $\pounds$ denote the liquid region, and let $\xi$ be the complex slope determining its complex structure. Kenyon–Okounkov conjecture. The random field

$$\sqrt{\pi}\big(\mathsf{Ht}_\mathcal N-\mathbb E[\mathsf{Ht}_\mathcal N]\big)$$

converges as $\mathcal N\to\infty$ in the liquid region $\pounds$ to the Gaussian free field with respect to the complex slope $\xi$ and with Dirichlet boundary conditions. This conjecture predicts the next-order stochastic fluctuations around the limit shape: frozen regions have no fluctuations, while the centered height function in the liquid region has universal Gaussian-free-field asymptotics. The supplied text does not state whether the conjecture has been resolved.