Gartland and Lokshtanov's induced grid minor conjecture

Given a graph $G$, a set $S\subseteq V(G)$ is a balanced separator if every component $C$ of $G-S$ satisfies $|V(C)|\le |V(G)|/2$. For sets $X,Y\subseteq V(G)$, say that $X$ dominates $Y$ if $Y\subseteq N[X]$.

Induced Grid Minor Conjecture. There exists a function $f$ such that for every planar graph $H$, every $H$-induced-minor-free graph $G$ has a balanced separator dominated by $f(H)$ vertices.

The source notes that the conjecture was originally stated for grid graphs and is equivalent to the planar-graph formulation because every planar graph is an induced minor of a sufficiently large grid. Its status is open.