Is the following, I call it bilinear Poincaré inequality,  true? 

Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in H^2(\Omega)$,
\begin{equation}
\int_{\Omega} (u - \bar{u})(v - \bar{v}) \dL x \le C\int_{\Omega} \lvert Du\rvert\lvert Dv\rvert\dL x
\end{equation}
where $\bar{u}$, $\bar{v}$ are the mean values of $u$ and $v$ over $\Omega$ respectively.