Jonsson's polytopality conjecture for multiassociahedra

Let $n$ and $k$ satisfy $n\geq 2k+1$, and let $\overline{\mathcal{A}\hskip-0pt\mathit{ss}}_{k}(n)$ be the reduced multiassociahedron, whose facets are reduced $k$-triangulations.

Jonsson's polytopality conjecture. For every $n\geq 2k+1$, the complex $\overline{\mathcal{A}\hskip-0pt\mathit{ss}}_{k}(n)$ is a polytopal sphere: there is a simplicial polytope of dimension $k(n-2k-1)-1$ with $\binom{n}2-kn$ vertices whose lattice of proper faces is isomorphic to $\overline{\mathcal{A}\hskip-0pt\mathit{ss}}_{k}(n)$.

Jonsson proved that the reduced multiassociahedron is a shellable sphere; the conjecture asks for a realization as the boundary complex of a simplicial polytope.