On a matrix partition conjecture

In 1977, Ganter and Teirlinck proved that any 2t × 2t matrix with 2t nonzero elements can be partitioned into four submatrices of order t of which at most two contain nonzero elements. In 1978, Kramer and Mesner conjectured that any mt × nt matrix with kt nonzero elements can be partitioned into mn submatrices of order t of which at most k contain nonzero elements. We show that this conjecture is true for some values of m, n, t and k but that it is false in general.