Grassmannian forest characterization of momentum-amplituhedron boundaries

Let $\mathcal{M}_{n,k}$ be the momentum amplituhedron. For a Grassmannian graph $G$, let $\Phi^\circ_G$ be the associated stratum, and let $\dim_{\mathcal{M}}(F)$ denote the dimension assigned to a Grassmannian forest $F$. Grassmannian forest boundary conjecture. The stratum $\Phi^\circ_G$ is a boundary of $\mathcal{M}_{n,k}$ if and only if $G$ is a Grassmannian forest of type $(k,n)$. Moreover, for every Grassmannian forest $F$,

$$\dim \Phi^\circ_F=\dim_{\mathcal{M}}(F).$$

This conjecture is motivated by boundary data for $4\leq n\leq 8$ and $2\leq k\leq n-2$, where the relevant positroid cells were observed to be labelled by Grassmannian forests; no general proof is given in the source.