The minimax theorem of von Neumann says that for any payoff matrix $A$, we have \begin{equation} \max_x \min_y x^T A y = \min_y \max_x x^T A y. \end{equation} In the above, $x$ and $y$ are probability distributions. In the special case when $A$ satisfies: $$\max_i \min_j A_{ij} = \min_j \max_i A_{i,j} (=A_{\bar{i},\bar{j}}),$$ the maximizing distribution $x$ and the minimizing $y$ in von Neumann's equality above is simply $x = e_{\bar{i}}$ and $y = e_{\bar{j}}$. (The Nash equilibrium is attained by pure strategies for both players.)

In a yet more (?) specific case when $A$ has a dominant column or a dominant row, i.e. there exists a $\bar{j}$ so that $$A_{i,\bar{j}} \leq A_{i,j}, \;\;\; \forall i, \; \forall j,$$ or there exists a $\bar{i}$ so that $$A_{\bar{i},{j}} \geq A_{i,j}, \;\;\; \forall i, \; \forall j,$$ it is quite easy to show that $A$ satisfies the second equality above.

Question: Does the second equality above imply the existence of a dominant row or column?