Polytopality conjecture for compatibility fans of graphical nested complexes

A graphical nested complex is associated with a graph, and its primal and dual compatibility fans are the fans constructed from the corresponding compatibility data. Polytopality conjecture. All primal and dual compatibility fans of graphical nested complexes are polytopal. This would extend the known polytopality of graphical nested fans, or graph associahedra, to the compatibility fans; the source states that polytopality remains open in the cases under consideration.