Homotopy type of complex-reflection-group noncrossing parking-function posets

Establish that for every well-generated complex reflection group W of rank n and Coxeter number h, the order complex of the proper part of the W-noncrossing parking-function poset is homotopy equivalent to a wedge of (h-1)^n spheres of dimension n-1.

Background

The paper studies noncrossing parking functions for irreducible well-generated complex reflection groups and constructs combinatorial and algebraic models that are equivariantly isomorphic for the groups treated. In the final section, the author turns to the topology of the associated noncrossing parking-function posets. For a finite poset with unique minimum and maximum, the proper part is obtained by removing those two elements, and its order complex is the simplicial complex of chains in the resulting poset.

Prior work cited in the paper establishes that, for finite real reflection groups, the order complex of the proper part of the noncrossing parking-function poset is homotopy equivalent to a wedge of (h-1)^n spheres of dimension n-1. The stated conjecture asks whether the same homotopy-type formula extends to well-generated complex reflection groups.