Extension of the double-dimer result to CP hyperbolic triangulations

Let $G$ be a CP hyperbolic triangulation, without necessarily assuming bounded degree or nonamenability. Extension conjecture. The statements of Theorems $\mathrm{thm:main0}$ and $\mathrm{thm:main}$ hold for CP hyperbolic triangulations. The authors expect that the bounded-degree and nonamenability assumptions can be removed, leaving CP hyperbolicity as the relevant condition; this extension is posed as an open question.