Gartland–Lokshtanov dominated balanced separator conjecture

Let $H$ be a planar graph. A graph $G$ is $H$-induced-minor-free if it does not contain $H$ as an induced minor. A set $S\subseteq V(G)$ is dominated by $C\subseteq V(G)$ if $S\subseteq N[C]$, and a balanced separator is a separator whose deletion leaves every component with at most half of the vertices of $G$.

Gartland–Lokshtanov's conjecture. For every planar graph $H$ there exists an integer $k_H$ such that every $H$-induced-minor-free graph $G$ has a balanced separator dominated by at most $k_H$ vertices.

This is proposed as an induced-minor analogue of the Grid Minor Theorem and would provide structural and algorithmic consequences for induced-minor-free graph classes. The supplied text gives no evidence that the conjecture has been resolved.