The wedge-of-spheres conjecture for noncrossing parking function posets

Let $W$ be a well-generated complex reflection group of rank $n$ and Coxeter number $h$. Let $\operatorname{Park}^{\mathrm{NC}}(W)$ denote its noncrossing parking function poset, let $\overline{\operatorname{Park}^{\mathrm{NC}}(W)}$ be its proper part, and let $\Omega$ denote the order complex.

Wedge-of-spheres conjecture.

$$\Omega\left(\overline{\operatorname{Park}^{\mathrm{NC}}(W)}\right)\simeq \left(\mathbb{S}^{n-1}\right)^{\vee (h-1)^n}.$$

This conjecturally extends the known result for finite real reflection groups to well-generated complex reflection groups, describing the topology of the noncrossing parking function poset as a wedge of spheres.