Benjamini--Schramm convergence for graphs excluding a finite set of minors

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.

Minor-exclusion Benjamini--Schramm conjecture. The sequence $G_n$ Benjamini--Schramm converges.

The conjecture is the unrooted counterpart of the rooted weak-convergence question. The paper proves it for several subcritical families, but the general minor-closed case is not established here.