Gartland–Lokastov induced-minor separator conjecture

Let $H$ be a planar graph. A graph is $H$-induced-minor-free if it does not contain $H$ as an induced minor, and a graph admits a $(k,1)$-balanced separator if, for every vertex-weight function, it has a $(k,1)$-centred set that is a balanced separator.

Gartland–Lokastov's conjecture. For every planar graph $H$, there exists $k\in\mathbb{N}$ such that every $H$-induced-minor-free graph $G$ admits a $(k,1)$-balanced separator.

This conjecture proposes that excluding a large grid as an induced minor is the only obstruction to balanced separators in hereditary graph classes. It is known for several classes, including $P_t$-free, even-hole-free, and three-path-configuration-free graphs, but remains open in general.