Weak convergence conjecture for bounded-degree planar triangulations

Let $c4_N$ denote the distribution of the relevant finite planar triangulations, conditioned to have all vertex degrees bounded by $M$. Weak convergence conjecture. For every $Mge 6$, the distributions $c4_N$ conditioned to have degrees uniformly bounded by $M$ are weakly convergent. This concerns existence of a limiting distribution for bounded-degree planar triangulations; the supplied text states it as a conjecture appearing in Benjamini and Schramm and gives no evidence of a resolution.