Homotopy type of double skewering complexes

Identify the homotopy type of double skewering complexes for quasi-median graphs and collections of gated subgraphs arising from graph products of finite groups.

Background

The finite-group coset intersection complexes can be interpreted as double skewering complexes, whose simplices are collections of gated subgraphs crossed by two non-transverse hyperplanes. Although this gives a geometric reformulation, the authors state that determining the homotopy type of such complexes remains unresolved.

References

However, identifying the homotopy type of a double skewering complex does not seem to be straightforward, and the problem remains to be investigated.

— 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 “Right-angled Coxeter groups”