Polytopal-subdivision realization conjecture for s-permutahedra

For each sequence $s$, let the $s$-permutahedron be the generalized permutahedral object defined in the source. A polytopal subdivision is a subdivision whose cells are polytopes. Polytopal-subdivision realization conjecture. A geometric realization of the $s$-permutahedron as a polytopal subdivision of the permutahedron exists in all dimensions.

The conjecture asks for a geometric realization simultaneously compatible with the permutahedron's polytopal subdivision structure. The source says that later flow-polytope work would partially solve it for sequences $s$ without zeros, while extension to the zero case remains a natural next step.