Singular-locus conjecture for dual periods of rational polypols

Let $\mathcal P$ be a regular rational polypol, and let $F_{\mathcal P}(u,v)$ denote its associated dual period. Consider the holonomic system annihilating $F_{\mathcal P}$, and the vertex hyperplanes and projective dual curves associated with the boundary components of $\mathcal P$. Singular-locus conjecture. The singular locus of the holonomic system annihilating $F_{\mathcal P}$ is contained in the union of the vertex hyperplanes and the projective duals of the nonlinear boundary components. If the boundary is sufficiently generic, all these components occur. This predicts that projective duality controls the singular support of the dual period, extending the polyhedral picture to rational polypols with curved boundary components. The surrounding discussion presents this as a problem for further study, and no resolution is given.