Armstrong's F=M conjecture for generalised non-crossing partitions

Let $\Phi$ be a finite root system of rank $n$, let $m$ be a positive integer, and let $F^m_\Phi(x,y)$ be the $F$-triangle of the generalised cluster complex. Let $NC^m(\Phi)$ be the associated generalised non-crossing partition poset, with M-triangle

$$M^m_\Phi(x,y)=\sum_{u,w\in NC^m(\Phi)}\mu(u,w)x^{\operatorname{rk}u}y^{\operatorname{rk}w}.$$

Write $(NC^m(\Phi))^*$ for its dual poset, with rank function $\operatorname{rk}^*$ and Möbius function $\mu^*$. Armstrong's $F=M$ conjecture. For every finite root system $\Phi$ of rank $n$,

$$F^m_\Phi(x,y)=y^nM^m_\Phi\left(\frac{1+y}{y-x},\frac{y-x}{y}\right).$$

Equivalently,

$$(1-xy)^nF^m_\Phi\left(\frac{x(1+y)}{1-xy},\frac{xy}{1-xy}\right)=\sum_{u,w\in(NC^m(\Phi))^*}\mu^*(u,w)(-x)^{\operatorname{rk}^*w}(-y)^{\operatorname{rk}^*u}.$$

This conjecture relates the refined face enumeration of generalised cluster complexes to Möbius-function enumeration in generalised non-crossing partition posets. The displayed identity was proved in the paper for type $D_n$, but the assertion for arbitrary finite root systems is not established here.