The finite-completion characterization for low-rank matrix patterns

Let $\Omega$ be a matrix-observation pattern, let $r$ be the target rank, and suppose that every row and column of $\Omega$ contains at least $r+1$ entries equal to $1$. A rank-$r$ matrix $X$ is generic if its entries are sufficiently general, and a rank-$r$ completion of the observations of $X$ at $\Omega$ is a rank-$r$ matrix agreeing with $X$ on those observed entries. Condition ii) of Theorem is the combinatorial condition stated in the source.

Finite-completion conjecture. Without loss of generality, let the number of $1$'s at every row and column of $\Omega$ be at least $r+1$. Then a generic rank-$r$ matrix $X$ observed at $\Omega$ has finitely many rank-$r$ completions if and only if $\Omega$ satisfies condition ii) of Theorem.

The conjecture proposes that, after excluding rows and columns with too few observations, condition ii) exactly characterizes finite completability. The source notes that uniquely completable patterns exist beyond the sufficient condition of Theorem, but reports no counterexample to this finite-completion characterization.