Universal edge fluctuation conjecture for dimer models

Let $\Omega$ be a natural domain with natural boundary values for a dimer model, without gas and quasifrozen points. Suppose the underlying bipartite graphs $G_n$ converge to $\Omega$ in a suitable sense. At a frozen-boundary point $z_0$, distinguish the six local regularity cases: a smooth point, first-order cusp, or first-order tacnode, each with either Hölder-continuous gradient or noncontinuous gradient up to the boundary from inside the liquid domain.

Universal Edge Fluctuation Conjecture. Under suitable assumptions on the convergence of $G_n$, all scaling limits of the determinantal point processes at the frozen boundary are given by the corresponding processes in the list: the extended Airy, Pearcey, Tacnode, GUE corner, Cusp-Airy, and discrete Tacnode processes, together with interpolation processes between these.

The listed processes are known for special natural domains and suitable assumptions. The conjecture asserts their universality under the stated convergence hypotheses, while the precise suitable assumptions and the general proof remain open.