Amplituhedra as positive geometries

Prove that for any integers k,n,m and any real n×(k+m) matrix Z with all maximal minors positive, the amplituhedron A_{k,n,m}(Z) (the image of the non-negative Grassmannian Gr_{k,n}^{≥0} under right multiplication by Z) is a positive geometry in the sense of Arkani-Hamed–Bai–Lam; that is, show that A_{k,n,m}(Z) admits a unique canonical differential form with only simple poles on its boundaries and residues equal to the canonical forms of the boundary components.

Background

The amplituhedron A_{k,n,m}(Z) is defined as the image of the positive Grassmannian under a linear map induced by a totally positive matrix Z. Positive geometries are semi-algebraic domains equipped with a uniquely characterized canonical form obeying recursive residue axioms on their boundaries.

While the positive Grassmannian is known to be a positive geometry, it remains conjectural that its images—amplituhedra—inherit this positive-geometry structure in full generality. Establishing this would give a firm mathematical foundation to a geometric formulation of scattering amplitudes and their canonical forms.