Polytope-complex conjecture for boundaries of hyperbolic posets

Let $F$ be a real-rooted polynomial, let $\mathcal{L}_s(F)$ be the lattice associated with its hyperbolic slice, and let $\mathcal{L}^\Delta_s(F)$ denote its dual; write $\partial(\mathcal{L}^\Delta_s(F))$ for the boundary complex of this dual. Polytope-complex conjecture. The boundary complex $\partial(\mathcal{L}^\Delta_s(F))$ is a polytope complex and hence is a combinatorial sphere. This is a weaker conjecture than the general polytopality conjecture; the paper notes that the claim is not known in general, while it holds in the generic setting through the established combinatorial-sphere results.