Bounded-degree enclosure conjecture for proper minor-closed classes

Let ${\mathcal C}_0$ be a proper minor-closed class of countable graphs. For a minor-closed graph family ${\mathcal C}$, let $\Delta({\mathcal C})$ be the smallest cardinal $k$ such that every $G\in{\mathcal C}$ has a graph $H\in{\mathcal C}$ with $\Delta(H)\leq k$ and $G$ is a minor of $H$. Bounded-degree enclosure conjecture. Every proper minor-closed class of countable graphs is contained in a minor-closed class ${\mathcal C}$ with finite $\Delta({\mathcal C})$. The question concerns which minor-closed classes have finite degree parameter; the source gives no resolution of this more specific problem.