Compactified weak convergence for rooted minor-excluding graphs

Let $\mathbb G^*$ be the compactification of rooted countable graphs introduced in the paper, with metric $d$. Let $\mathcal S$ be a finite set of finite connected graphs, and let $Ex^{\bullet}(\mathcal S)$ be the class of rooted, labelled or unlabelled, connected graphs with no minor in $\mathcal S$. Let $R_n$ be a uniformly random element of $Ex^{\bullet}(\mathcal S)$ with $n$ vertices.

Compactified rooted convergence conjecture. The sequence $R_n$ converges weakly with respect to $d$.

The compactification allows infinite-degree vertices and is compact, so every sequence has a convergent subsequence. The conjecture asserts convergence of the entire sequence for minor-excluding classes; it is closely related to the earlier rooted weak-convergence formulation but is not merely ordinary weak convergence.