Martin boundary conjecture for reversible random planar maps

Let $(M,\rho)$ be a reversible random planar map. The Martin boundary of $M$ is the boundary describing asymptotic behavior of positive harmonic functions for random walk on $M$. A map is invariantly amenable or invariantly non-amenable according to its invariant expansion, and it may be one-ended or infinitely-ended. Martin boundary conjecture for reversible random planar maps. Almost surely, the Martin boundary of $M$ is homeomorphic to one of the following: a point if $M$ is invariantly amenable; the circle $\partial\mathbb D$ if $M$ is invariantly non-amenable and one-ended; or the space of ends of $M$ if $M$ is invariantly non-amenable and infinitely-ended. In the second case, the homeomorphism should arise from circle packing a suitable triangulation associated to $M$. This would strengthen the identification of geometric and Poisson boundaries to the Martin boundary. The cited bounded-degree case is known, while the stated general reversible-map version is left as a belief in the paper.