The Erdős–Pósa parameter conjecture for minor-closed graph classes

Let $H$ be a graph. A graph class is minor-closed if it contains every minor of each of its graphs. A graph parameter is minor-monotone if its value does not increase when taking a minor. For a graph class $\mathcal{G}$, a parameter is bounded in $\mathcal{G}$ if there is some $c oindentin\mathbb{N}$ such that its value is at most $c$ on every graph in $\mathcal{G}$. Let $\mathcal{M}_H$ denote the class of graphs containing $H$ as a minor.

Erdős–Pósa parameter conjecture. For every graph $H$, there exists a minor-monotone graph parameter $\text{\scriptsize\textsf{EP}}_H$ such that $\mathcal{M}_H$ has the Erdős–Pósa property in a minor-closed graph class $\mathcal{G}$ if and only if $\text{\scriptsize\textsf{EP}}_H$ is bounded in $\mathcal{G}$.

For planar $H$, the constant zero function gives such a parameter, but existence for non-planar $H$ is not known. The assertion is tied to the open conjecture that graphs are $\omega^2$-well-quasi-ordered by minors, and a constructive or canonical description of the parameter would have algorithmic significance.