Finiteness of fibers in matrix completion via Plücker coordinates

Let $\Omega \subseteq \{1,\dots,m\} \times \{1,\dots,n\}$. We consider fibers of coordinate projections $\pi_\Omega : \mathscr{M}_k(r,m \times n) \rightarrow k^{\# \Omega}$ from the algebraic variety of $m \times n$ matrices of rank at most $r$ over an infinite field $k$. For $\#\Omega = \dim \mathscr{M}_k(r,m \times n)$ we describe a class of $\Omega$'s for which there exist non-empty Zariski open sets $\mathscr{U}_\Omega \subset \mathscr{M}_k(r,m \times n)$ such that $\pi_\Omega^{-1}\big(\pi_\Omega(X)\big) \cap \mathscr{U}_\Omega$ is a finite set $\forall X \in \mathscr{U}_\Omega$. For this we interpret matrix completion from a point of view of hyperplane sections on the Grassmannian $\operatorname{Gr}(r,m)$. Crucial is a description by Sturmfels $\&$ Zelevinsky of classes of local coordinates on $\operatorname{Gr}(r,m)$ induced by vertices of the Newton polytope of the product of maximal minors of an $m \times (m-r)$ matrix of variables.