Benjamini--Schramm convergence to the UIPT for maximal surface-embeddable graphs

Fix a surface. A graph is edge-maximal with respect to embeddability in the surface if it is embeddable in that surface, but adding any edge makes it non-embeddable. Let $G_n$ be a uniformly random labelled or unlabelled graph with $n$ vertices having this property. The UIPT is the uniform infinite planar triangulation.

Maximal surface-graph conjecture. The sequence $G_n$ Benjamini--Schramm converges, and its limit coincides with the UIPT.

This conjecture is motivated by seeking random-geometry limits more interesting than the tree-like limits arising for subcritical classes. The source presents it as a possibility for maximal graphs and does not report a proof.