Quasi-isometric invariance and homotopy type of the small crossing graph

Determine whether the small crossing graph of the quasi-median graph associated to a graph product is preserved by quasi-isometries and identify the homotopy type of its flag completion.

Background

The small crossing graph was introduced for quasi-median graphs associated to graph products and is known in the cited setting to admit a natural automorphism-group action. The paper leaves unresolved both its preservation under quasi-isometries and the homotopy type of the flag completion of that graph.

References

Is the small crossing graph also preserved by quasi-isometries? Can we identify the homotopy type of its flag completion?

— Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups  (2503.08411 - Abbott et al., 11 Mar 2025) in Section 6, paragraph “Other quasi-isometric invariants”