The degree-one characterization conjecture for amplituhedron cells

Let $f\in\tilde{S_n}(-k,n-k)$ be an affine permutation with $\operatorname{inv}(f)=k\ell$. A cell $f$ is $(n,k,m)$-admissible when it satisfies the source's admissibility condition; it has degree $1$ when the corresponding rational map is birational for generic $Z$, and has $Z$-degree $1$ when this holds for the specified $Z$. Degree-one characterization conjecture. The following are equivalent:

1. $f$ is $(n,k,m)$-admissible.
2. $f$ has degree $1$.
3. $f$ has $Z$-degree $1$ for every $Z\in\operatorname{Gr}_{>0}(k+m,n)$.

In each case, $Z$ is a diffeomorphism from $\Pi^{>0}_{k+f}$ to $Z(\Pi^{>0}_{k+f})$. This conjecture seeks an intrinsic characterization of the cells that map birationally to amplituhedron cells; the source gives no resolution.