Positive-geometry property of all positive hexahedra

Determine whether every positive hexahedron in \(\mathrm{Gr}(2,4)\) is a positive geometry, including whether the residue conditions at its boundary vertices select a unique numerator of the canonical form even when multiple adjoint hypersurfaces exist.

Background

The paper proves that positive pentahedra satisfying the stated hypotheses are positive geometries and verifies the positive-geometry property for several positive hexahedra. For hexahedra, however, the adjoint need not be unique: the paper exhibits an example with a three-dimensional family of adjoints. This motivates the unresolved question of whether the canonical-form residue conditions nevertheless determine a unique numerator for every positive hexahedron.

References

Are positive hexahedra in $\mathrm{Gr}(2,4)$ positive geometries? While there may be many adjoints, the residue conditions at the vertices of the hexahedron might still pick a unique one as the numerator of the canonical function.

— Positive Polytopes with Few Facets in the Grassmannian  (2503.01652 - Pavlov et al., 3 Mar 2025) in Question environment, Section 6, “Open questions and future directions”