It is well-known that the following Hardy inequality holds: \begin{equation} \frac{(N-2)^2}{4}\int_{\mathbb{R}^N}\frac{u^2}{|x|^2}\,dx\leq\int_{\mathbb{R}^N}|\nabla u|^2\,dx,\quad\quad\forall\,u\in\mathcal{D}^{1,2}(\mathbb{R}^N), \end{equation} where $\mathcal{D}^{1,2}(\mathbb{R}^N)$ is the completion of $C_0^\infty(\mathbb{R}^N)$ under the norm $\|u\|^2=\int\,|\nabla u|^2$.
My question is that is there exist the reverse type inequality of the above inequality? Or, equivalently, whether the following proposition holds for some constant $C>0$:
\begin{equation} \int_{\mathbb{R}^N}|\nabla u|^2\,dx\leq C\int_{\mathbb{R}^N}\frac{u^2}{|x|^2}\,dx,\quad\quad\forall\,u\in\mathcal{D}^{1,2}(\mathbb{R}^N). \end{equation}
I guess this conjecture is wrong, but I can't construct a counter-example. Any hints will be appreciated!