Shellability and homotopy conjecture for positive generalized cluster complexes

Let $\Phi$ be a crystallographic root system of rank $\ell$, let $m\geq 1$, and let $\Delta^m_+(\Phi)$ be the positive part of the generalized cluster complex, with $N^+(\Phi,m-1)$ denoting the number of bounded dominant regions of the corresponding generalized Catalan arrangement for parameter $m-1$. Positive cluster-complex shellability conjecture. The complex $\Delta^m_+(\Phi)$ is pure of dimension $\ell-1$ and shellable, and

$$\widetilde{\chi}(\Delta^m_+(\Phi))=(-1)^{\ell-1}N^+(\Phi,m-1).$$

In particular, it is Cohen--Macaulay and has the homotopy type of a wedge of $N^+(\Phi,m-1)$ spheres of dimension $\ell-1$. This is stated as the positive analogue of a conjecture of Fomin and Reading; the supplied text gives no resolution status.