Polytopal realizations of spherical subword complex dual graphs

Establish whether the dual graphs of spherical subword complexes admit realizations as the 1-skeletons of convex polytopes by constructing such polytopal realizations or proving that no such realization exists.

Background

The discussion connects associahedra and various generalizations, noting that shortest-path problems on these structures can be NP-hard.

For spherical subword complexes, the authors explicitly point out a long-standing conjecture about polytopal realizations and that no realization is currently known.