Induced-minor-free product structure conjecture

Let $H$ be a graph. For a graph $G$, being $H$-induced-minor-free means that $G$ does not contain $H$ as an induced minor. Let $J\boxtimes P$ denote the strong product of graphs $J$ and $P$, and let $\operatorname{tw}(J)$ denote the treewidth of $J$.

Induced-minor-free product structure conjecture. There is a function $f\colon \mathbb{N}\to\mathbb{N}$ such that every $H$-induced-minor-free graph $G$ with maximum degree at most $\Delta$ is isomorphic to a subgraph of $J\boxtimes P$ for some graph $J$ with $\operatorname{tw}(J)\leqslant f(\Delta)$ and for some path $P$.

This is proposed as a common strengthening of product-structure results for planar and bounded-degree minor-free graphs. Its resolution is not given in the source.