The flag-polytope characterization of TNS toric manifolds

Let $M^{2n}$ be a nonsingular projective toric variety of complex dimension $n$, and let $P^n$ be its moment polytope. Flag-polytope characterization conjecture. $M^{2n}$ is a TNS-manifold if and only if $P^n$ is a flag polytope. The preceding theorem proves that a TNS toric manifold has no minimal missing faces on $n$ vertices and, in particular, has a flag moment polytope in dimension three. The converse and the stated characterization in arbitrary dimension remain open.