Conjecture on the limiting degree distribution of random planar graphs

For each integer $k\geq 1$, let $Z^{(k)}_n$ denote the proportion of vertices of degree $k$ in a uniformly random planar graph with $n$ vertices. Limiting degree-distribution conjecture. As $n\to\infty$, the random variable $Z^{(k)}_n$ converges in law to an explicit constant $\pi^{(k)}$, and these constants satisfy

$$\sum_k\pi^{(k)}=1.$$

This conjecture proposes a limiting probability distribution for vertex degrees in uniformly random planar graphs; the stated passage does not indicate that it has been resolved.