Weak-limit conjecture for maximal Schnyder woods of the UIHPT

Let the UIHPT denote the uniform infinite half-plane triangulation, and let $\cT_n^m$ be the class of triangulations used in the finite approximation, with a uniformly random triangulation chosen from $\cT_n^m$. Consider the unique maximal Schnyder wood of the UIHPT and the maximal Schnyder wood of the finite random triangulation.

Weak-limit conjecture. The unique maximal Schnyder wood of the UIHPT is the weak limit of the maximal Schnyder wood of a uniformly random triangulation from $\cT_n^m$ as $n \to \infty$ and then $m \to \infty$.

Establishing this conjecture would connect the infinite Schnyder wood of the UIHPT to maximal Schnyder woods on large finite uniformly random triangulations, potentially clarifying the limiting structure of geodesic paths and the winding behaviour of monochromatic trees. The source discusses these consequences but does not provide a proof.