Linear extension complexity conjecture for spanning tree polytopes of proper minor-closed graph families

Let $\mathcal{C}$ be a proper minor-closed family of graphs, and let $G=(V,E)$ be a connected graph in $\mathcal{C}$. Let $\operatorname{P_{\mathrm{sp.trees}}}(G)$ denote the convex hull of the characteristic vectors of the spanning trees of $G$. The minor-closed spanning tree polytope conjecture.

$$\operatorname{xc}(\operatorname{P_{\mathrm{sp.trees}}}(G))=O(|V|).$$

This would extend the proposed linear bound from fixed-surface graph families to all proper minor-closed families. The source presents it as a further possible generalization, and it remains open.