Interval matrices: realization of ranks by rational matrices

Let $\alpha$ be a $p \times q$ interval matrix with $p \geq q$ and with the endpoints of all its entries in the set of the rational numbers. We prove that, if $\alpha$ contains a rank-$r$ real matrix with $r \in \{2, q-2,q-1,q\}$, then it contains a rank-$r$ rational matrix.