Commuting nilpotent operators with maximal rank

Let $X$, $\hat{X}$ be commuting nilpotent matrices over a field $k$ with nilpotency $p^t$. We show that if $X-\hat{X}$ is a certain linear combination of products of commuting nilpotent matrices, then $X$ is of maximal rank if and only if $\hat{X}$ is of maximal rank. In the case, $k$ is an algebraically closed field of positive characteristic $p$ there is an interpretation of this result for modules over group algebras.