Sign patterns of rational matrices with large rank

Let AA be a real matrix. The term rank of AA is the smallest number tt of lines (that is, rows or columns) needed to cover all the nonzero entries of AA. We prove a conjecture of Li et al. stating that, if the rank of AA exceeds t−3t−3, there is a rational matrix with the same sign pattern and rank as those of AA. We point out a connection of the problem discussed with the Kapranov rank function of tropical matrices, and we show that the statement fails to hold in general if the rank of AA does not exceed t−3t−3.