The injectivity and birationality criterion for positroid cells

Let $S_M$ be a $km$-dimensional positroid cell of ${\rm Gr}_+(k,n)$. Let $\pi_+$ be the positive amplituhedron map and let $\pi$ denote its complexified rational map on the closure $\overline{S_M}$. The injectivity criterion. The following conditions are equivalent: (i) the restriction

$$\pi_+:S_M\longrightarrow\pi_+(S_M)$$

is injective; and (ii) $\pi:\overline{S_M}\dashrightarrow {\rm Gr}(k,\mathbb{C}^{k+m})$ is birational and

\textup{\dim}(\overline{S_M})=\textup{\dim}(\pi(\overline{S_M}))=mk.

In either case, the restriction of $\pi_+$ is a diffeomorphism. The source places this assertion in a conjecture environment but gives no status evidence or discussion establishing it, so it is recorded as open pending verification.