Georgakopoulos–Papasoglu asymptotic-grid conjecture

Prove that if a graph excludes a k×k grid as an asymptotic minor for some integer k, then the graph is quasi-isometric to a graph of bounded treewidth.

Background

This conjecture is attributed to Georgakopoulos and Papasoglu. The paper explains that its validity would imply positive answers to the bounded-treewidth characterization question and to the excluded-planar-asymptotic-minor question, as well as the 1-or-∞ dichotomy for weak cop numbers in locally finite quasi-transitive graphs. The paper cites partial progress by Albrechtsen and Hamann.

References

If there is an integer $k$ such that a graph $G$ excludes the $k\times k$ grid as an asymptotic minor, then $G$ is quasi-isometric to a graph of bounded treewidth.

— Coarse cops and robber in graphs and groups  (2502.15571 - Esperet et al., 21 Feb 2025) in Conjecture 3.8, Section 3.2, subsection “Graphs quasi-isometric to graphs of bounded treewidth”