Kenyon–Okounkov Gaussian Free Field conjecture for lozenge tilings

Fix an arbitrary tilable polygonal domain $\Upsilon$ on the triangular grid. Let $H_L(x,y)$ be the random height function of a uniformly random lozenge tiling of the dilated domain $L\backslashcdot \Upsilon$; if $\Upsilon$ is not simply connected, fix the heights of its holes deterministically. Let $\mathcal L$ be the liquid region, let $xi$ be its complex slope, and let the Gaussian Free Field there be the centered Gaussian generalized field with covariance given by the Dirichlet Green function for the induced complex structure. Kenyon–Okounkov's lozenge-tiling conjecture. The random field

$$\sqrt{\pi}\left[H_L(x,y)-\mathbb E H_L(x,y)\right]$$

converges as $L\to\infty$ in $\mathcal L$ to the Gaussian Free Field with respect to the complex slope $xi$ and with Dirichlet boundary conditions. The source notes that this has been established for several multiply? No: the cited established examples are simply connected, so the arbitrary-domain assertion, including nonsimply connected domains, remains open.