Commuting Nilpotent Operators and Maximal Rank

Let X, [(X)tilde]{widetilde X} be commuting nilpotent matrices over k with nilpotency p ^t , where k is an algebraically closed field of positive characteristic p. We show that if X- [(X)tilde]{X- widetilde X} is a certain linear combination of products of pairwise commuting nilpotent matrices, then X is of maximal rank if and only if [(X)tilde]{widetilde X} is of maximal rank.