Exact diagonal-entry change bound for prescribed rank reduction

Determine whether, for every pair of integers m,k≤n and every matrix M∈Z₂^{n×n} of rank m, whenever a matrix of rank k can be obtained from M by changing entries on the main diagonal, a rank-k matrix can be obtained by changing exactly |m−k| diagonal entries.

Background

The paper defines R(M) as the minimum rank obtainable from a square matrix M over Z₂ by changing entries on its main diagonal. It proves that changing diagonal entries to transform a matrix of rank n into one of rank k requires at least |n−k| changes. The open problem asks whether this lower bound is always attainable whenever the target rank is achievable at all, thereby determining whether the basic rank-change estimate is sharp in every case.