Polynomial degree-distribution conjecture for random triangulations

Let $n$ denote the size of a triangulation, and let the average number of its nodes of degree $d$ be divided by $n$, where $d$ is an integer larger than $2$. Polynomial degree-distribution conjecture. There is a probability measure $p$ on the integers larger than $2$ such that the average number of nodes of degree $d$ divided by $n$ converges, as $n$ tends to infinity, to $p(d)$. Moreover, $p$ has polynomial decay in the sense that

$$d^{-4}p(d)$$

converges to a nonzero finite limit as $d$ tends to infinity. This conjecture formalizes the numerical observation that the model has a limiting power-law degree distribution, with the degree distribution behaving like $d^{-4}$. The paper reports numerical evidence but no theoretical explanation or proof.