Coxeter-generalization: homotopy equivalence of the branched annulus complex to the orbit configuration space

Show that for any Coxeter group W and Coxeter element δ ∈ W, the bisimplicial complex Br_{W,δ}^{m}(closed annulus) is homotopy equivalent to its interior, which is homeomorphic to the orbit configuration space Y_W.

Background

The authors extend their polynomial constructions from the symmetric group case to general Coxeter groups by introducing a bisimplicial complex Br_{W,δ}^{m}(closed annulus), built from algebraic labels related to Coxeter factorizations. They conjecture that this compactification captures the homotopy type of the orbit configuration space Y_W.

If proven, this would bridge the dual Artin group A_{W,δ}^* and the standard Artin group A_W, and—under additional compactification hypotheses—could imply their isomorphism, advancing the K(π,1) program for Artin groups.