Polytopality conjecture for hyperbolic posets

Let $F$ be a real-rooted polynomial and let $\mathcal{H}_s(F)$ be its hyperbolic slice, stratified by root multiplicity compositions; the associated hyperbolic poset records the incidence relations among these strata. Polytopality conjecture. Hyperbolic posets are polytopal, meaning that each hyperbolic poset is isomorphic to the face lattice of a polytope. The examples suggest this because the strata have geometric and combinatorial properties reminiscent of polytopes, although the strata themselves need not be convex; the paper leaves the question open.