Given an infinite planar graph $G$, let's denote by $\{H_1,H_2,\dots,H_m\}$ all the labeled graphs on $n$ vertices that appear as subgraphs of $G$. Also let $$d_n=\frac{\sum_{i=1}^m \#E(H_i)}{nm}$$ denote the "edge density" among the $H$'s.

**Question:** Does the limit $\lim_{n\to \infty}d_n$ always exist? If yes, what does it say about the graph $G$?

The analogous question when we consider arbitrary infinite graphs, and define $d_n$ as $\frac{\sum_{i=1}^m \#E(H_i)}{\binom{n}{2}m}$, was asked by Peter Cameron here (BCC12.5), in an attempt to define a notion of "edge density" for infinite graphs. As far as I know this version is open, but I was hoping the planar case might be simpler.