The induced-minor coarse grid conjecture

Let $n\in\mathbb{N}$. An induced $(n\times n)$-grid minor is the indicated induced-minor model, and $(M,A)$-quasi-isometry and tree-width have their usual meanings. Then there exist some $M,A,g\in\mathbb{N}$ such that every graph with no $(n\times n)$-grid induced minor is $(M,A)$-quasi-isometric to a graph of tree-width at most $g$. Induced-minor coarse grid conjecture.

This is proposed as a possible weakening of the refuted Coarse Grid Theorem; the paper notes a bounded-degree theorem of Korhonen in this direction, while the unrestricted assertion remains open.