The Z-independence conjecture for amplituhedron combinatorics

Fix $Z i\operatorname{Gr}_{>0}(k+m,n)$. Let $f,f_1,\dots,f_N$ be affine permutations in $\tilde{S_n}(-k,n-k)$, and let $g$ be another such affine permutation. The notions of $Z$-admissibility, $Z$-compatibility, and $Z$-triangulation are defined for these affine permutations. Z-independence conjecture. The following equivalences hold:

1. $f$ is $Z$-admissible if and only if $f$ is $(n,k,m)$-admissible.
2. $f$ and $g$ are $Z$-compatible if and only if they are $(n,k,m)$-compatible.
3. $f_1,\dots,f_N$ form a $Z$-triangulation if and only if they form an $(n,k,m)$-triangulation.

This would make the combinatorics independent of the positive Grassmannian point $Z$. The source says it is widely believed in the physics literature, but gives no proof or resolution.