Homotopy type of the small crossing complex

Determine the homotopy type of the small crossing complex of an arbitrary quasi-median graph.

Background

The small crossing complex is defined using maximal hyperplanes of a quasi-median graph, with simplices corresponding to pairwise transverse maximal hyperplanes. The authors observe that it is homotopy equivalent to a complement in the prism-completion, but leave open whether this description can be simplified and, more fundamentally, what its homotopy type is.

References

Given a quasi-median graph X, what is the homotopy type of its small crossing complex \mathrm{sCross}\triangle(X)?

— 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,” Question