A more rigid homotopy-equivalence notion

Construct a more rigid notion of homotopy equivalence whose invariant can be identified for relative contact complexes of quasi-median graphs while retaining more information than ordinary homotopy equivalence and remaining easier to determine than simplicial isomorphism.

Background

The paper obtains genuine isomorphisms between certain coset intersection complexes but identifies their homotopy types because homotopy equivalence is substantially easier to analyze. The authors ask whether an intermediate equivalence relation can preserve more geometric or combinatorial information than homotopy equivalence without requiring full isomorphism.

References

Is there a “more rigid” notion of homotopy equivalence such that the rigid homotopy equivalence type of relative contact complexes of quasi-median graphs can be identified?

— 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 “Rigid homotopy equivalence,” Question