Projective-join characterization of defect polytopes

Let $P$ be an integral polytope. A polytope is called a defect polytope when its associated projective toric variety has degenerate dual variety, equivalently when the invariant $c(P)$ vanishes. Projective-join conjecture. $P$ is a defect polytope if and only if it is a projective join of some kind. The paper reports that no examples with $c(P)=0$ outside the projective-join construction were found. The characterization is presented as an open conjecture.