Positive $h$-vector conjecture for generalized cluster complexes

Let $\Phi$ be an irreducible crystallographic root system of rank $\ell$, let $m\geq 1$, and let ${\mathcal A}^m_\Phi$ be its generalized Catalan hyperplane arrangement. Let $h^+_i(\Phi,m)$ count bounded dominant regions according to the number of walls of the form $(\alpha,x)=m$ that do not separate the region from the fundamental alcove, and let $h_i(\Delta^m_+(\Phi))$ be the $i$th entry of the $h$-vector of the positive part of the generalized cluster complex. Positive $h$-vector conjecture. For every $0\leq i\leq\ell$,

$$h^+_i(\Phi,m)=h_i(\Delta^m_+(\Phi)).$$

This conjecture is the positive analogue of a conjecture of Fomin and Reading; the supplied text gives no resolution status.