Weak convergence for rooted graphs excluding a finite set of minors

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.

Rooted minor-exclusion convergence conjecture. The sequence $R_n$ converges weakly.

This is an early formulation of the paper's main question. The paper proves the analogous convergence for important special cases, including subcritical classes, while the general minor-closed case remains open in this formulation.