Gartland and Lokshtanov's induced-minor-free graph algorithm conjecture

Let $H$ be a planar graph. For a fixed integer $r$ and fixed CMSO$_2$ formula $\psi$, $(\operatorname{tw} \leq r,\psi)$-MWIS is the problem of finding a maximum-weight vertex set inducing a graph of treewidth at most $r$ that satisfies $\psi$, or reporting that no such set exists. Also, $\operatorname{lcol}(3)$ denotes the relevant 3-color list-colouring problem. A graph is $H$-induced-minor-free if it does not contain $H$ as an induced minor.

Gartland and Lokshtanov's conjecture. For every fixed $r$ and CMSO$_2$ formula $\psi$, $(\operatorname{tw} \leq r,\psi)$-MWIS and $\operatorname{lcol}(3)$ can be solved in polynomial time in $H$-induced-minor-free graphs.

This conjecture extends polynomial-time algorithmic results from graphs excluding linear forests to induced-minor-free graphs for planar forbidden graphs. Its status is not established in the supplied text.