Generic polygon invariance conjecture for higher associahedra

Let $\mathcal{A}$ be the set of vertices of a generic convex $n$-gon, and let $\Upsigma_{\mathcal{A},k}$ be the associated higher associahedron. The generic polygon invariance conjecture. The $f$-vector of $\Upsigma_{\mathcal{A},k}$ depends only on $n$ and $k$. This would extend the combinatorial invariance of the ordinary associahedron to higher associahedra for generic convex polygons, despite the dependence on the point configuration in the nongeneric setting. The claim is supported in the source by computational evidence, but no resolution is given.