Height characterization of dependence in the determinantal matroid

Prove that a set \Omega\subseteq[m]\times[n] is dependent in the determinantal matroid M(r,[m]\times[n]) if and only if there exists T\subseteq[m]\times[n] such that \operatorname{height}(I_T)>|T\setminus\Omega|.

Background

Theorem 3.3 gives a sufficient height inequality for detecting dependence: if the height of the ideal generated by determinantal minors supported on T exceeds the number of entries of T omitted from \Omega, then \Omega is dependent. The paper observes that every dependent set in its examples is detected in this way.

The authors conjecture that this sufficient condition is also necessary, which would provide a complete characterization of all dependent sets in the determinantal matroid. They prove afterward that the conjecture holds for ranks r=1 and r\geq\min(m,n)-2.