The Fomin–Zelevinsky enumerator relation for antichains

Let $\Phi$ be a finite root system of rank $n$. Define

$$H(x,y)=\sum_{k,\ell}h_{k,\ell}x^k y^\ell,$$

where $h_{k,\ell}$ counts antichains of cardinality $k$ containing $\ell$ simple roots. Let $F(x,y)=\sum_{k=0}^{n}\sum_{\ell=0}^{n}f_{k,\ell}x^k y^\ell$, where $f_{k,\ell}$ counts the simplices of the generalized associahedral complex having exactly $k$ positive-root vertices and $\ell$ negative-simple-root vertices. Enumerative relation conjecture. One should have

$$H(x,y)=(1-x)^n F\left(\frac{x}{1-x},\frac{xy}{1-x}\right).$$

The relation would connect the two-parameter antichain enumerator with the face enumerator of the generalized associahedron. The supplied text gives no resolution status.