Compactified local convergence for unrooted minor-excluding graphs

Let $\mathcal S$ be a finite set of finite connected graphs, and let $Ex(\mathcal S)$ be the class of labelled or unlabelled connected graphs with no minor in $\mathcal S$. Let $G_n$ be a uniformly random element of $Ex(\mathcal S)$ with $n$ vertices, with a root chosen uniformly at random from its vertices. Regard the resulting rooted graph as an element of the compactified space with metric $d$.

Compactified unrooted convergence conjecture. The sequence $G_n$ converges weakly with respect to $d$.

This is the compactified analogue of Benjamini--Schramm convergence and is designed to handle unbounded degrees near the random root. The source gives no resolution of the general conjecture.