Are amplituhedra positive geometries?

Determine whether, for arbitrary parameters k, m, and n and any totally positive matrix Z of size n × (k + m), the tree-level amplituhedron A_{n,k,m}(Z), defined as the image of the non-negative Grassmannian Gr(k, n)_{≥0} under the linear map V ↦ VZ, is a positive geometry in the sense of Arkani-Hamed and Lam.

Background

Amplituhedra are semialgebraic sets defined as images of the non-negative Grassmannian under linear maps specified by totally positive matrices, introduced in the context of scattering amplitudes. Positive geometries are a class of spaces equipped with canonical differential forms, and establishing that a given amplituhedron is a positive geometry links geometric structure with the computation of scattering amplitudes.

It is known that polytopes are positive geometries, and there are special cases where certain amplituhedra are known to be positive geometries (e.g., k = m = 2). The present paper proves that the limit amplituhedron for m = 2 (obtained by sending the number of particles n to infinity) is a positive geometry. However, the general question of whether amplituhedra are positive geometries remains unresolved.