Wachspress conjecture for regular polypols

Prove that the adjoint curve of every regular polypol does not intersect the interior of its associated semi-algebraic set.

Background

For a regular polypol, the boundary contains no singular points of the total boundary curve except the designated vertices, and the boundary is disjoint from the interior of the associated semi-algebraic set. The adjoint curve is the algebraic curve determined by the residual arrangement. The stated conjecture would ensure that this adjoint curve does not enter the region represented by the polypol, a property relevant to the construction and geometric interpretation of canonical forms. The notes explicitly describe the conjecture as wide open.