I think the statement is even true for $s=mn-n+2$. Let $w_i$ be the sum of entries of $v_i$. Then clearly $n\leq w_1\leq\dots\leq w_s\leq mn$, hence for $s=mn-n+2$ we have $w_i=w_j$ for some $i<j$. However $v_i\leq v_j$, so we must have $v_i=v_j$ (otherwise we would have $w_i<w_j$).