Functional monotonicity conjecture for contraction distance to minor-closed classes

Let $\mathcal{H}$ be a minor-closed graph class. For a graph $G$, let $\mathsf{id}_{\mathcal{H}}(G)$ denote the minimum number of edge identifications needed to transform $G$ into a graph in $\mathcal{H}$, and let $\mathsf{ec}_{\mathcal{H}}(G)$ denote the minimum number of edge contractions needed to transform $G$ into a graph in $\mathcal{H}$. Functional monotonicity conjecture. There is a function $f_{\mathcal{H}}:\mathbb{N}\to\mathbb{N}$ such that for every graph $G$,

$$\mathsf{ec}_{\mathcal{H}}(G)\leq f_{\mathcal{H}}(\mathsf{id}_{\mathcal{H}}(G)).$$

This would extend the displayed polynomial relationship proved in the paper for forests to every minor-closed class, showing that edge-contraction distance is bounded as a function of edge-identification distance; no resolution is given.