Minimal-rank completion conjecture for bipartite chordal patterns

Let a partial matrix have a bipartite graph whose vertices represent its rows and columns and whose edges represent specified entries. A fully specified submatrix is one in which every entry is specified, and a completion of rank $r$ is a matrix agreeing with all specified entries and having rank $r$. Minimal-rank completion conjecture. Suppose the bipartite graph is bipartite chordal, every fully specified submatrix has full rank, and every $k\times k$ submatrix has at most $(2k-r)r$ specified entries. Then there exists a completion of rank $r$. The conjecture proposes that bipartite chordality together with the stated density and nondegeneracy conditions is sufficient for a rank-$r$ completion; the preceding discussion establishes the result for $r=1$ and gives a counterexample without the bipartite-chordal hypothesis for $r\geq 2$.