Chapoton's H-triangle identity for noncrystallographic root systems

Let $\Phi$ be a noncrystallographic root system of rank $n$. Define the antichain generating function

$$H(\Phi;s,t)=\sum_{A\in\mathcal{A}(\Phi)}s^{|A\cap\Delta|}t^{|A|}.$$

Let $N\!C(\Phi)$ be the noncrossing partition lattice and let

$$M(\Phi;x,y)=\sum_{\substack{\sigma,\pi\in N\!C(\Phi)\sigma\leq\pi}}\mu(\sigma,\pi)x^{n-\operatorname{rk}(\sigma)}y^{n-\operatorname{rk}(\pi)}$$

be its Möbius generating function. Chapoton's H-triangle identity. Chapoton's $H$-triangle is related to the $M$-triangle by

$$H(\Phi;s,t)=(1+(s-1)t)^nM\left(\Phi;\frac{s}{s-1},\frac{(s-1)t}{1+(s-1)t}\right).$$

This conjecture asserts a numerical connection between antichains in the root poset and the Möbius function of the noncrossing partition lattice; the source states that it remains open.