Sphere dimension and quasi-isometric bounded sphere dimension for induced-minor-free graphs

Determine whether, for every finite graph H, all graphs in the class of graphs excluding H as an induced minor have bounded sphere dimension; if not, determine whether every such graph is uniformly quasi-isometric to a graph with bounded sphere dimension.

Background

For a finite graph H, the paper defines Forb_ind(H) as the class of graphs that do not contain H as an induced minor. It asks whether this exclusion condition forces bounded sphere dimension. The alternative question concerns a weaker coarse-geometric property: uniform quasi-isometry to graphs whose sphere dimension is bounded. Both possibilities are explicitly posed as unresolved problems.

References

Do the graphs in $\mathrm{Forb_{ind}(H)$ have bounded sphere dimension for every finite graph $H$? If not, is each graph in $\mathrm{Forb_{ind}(H)$ uniformly quasi-isometric to a graph with bounded sphere dimension?

— Strongly sublinear separators and bounded asymptotic dimension for sphere intersection graphs  (2504.00932 - Davies et al., 1 Apr 2025) in Section 6, Problem labeled \cref{prob forb H}