Boundedness of $L^2$-norms of mixed derivatives of functions from $W^{2,2}(\mathbb{R}^N)$

Suppose that we have a sequence of functions $u_n \in W^{2,2}(\mathbb{R}^N)$, $N \geq 3$, such that
\|\nabla u_n\|_{L^2(\mathbb{R}^N)} = 1
\quad \text{and} \quad
\|\Delta u_n\|_{L^2(\mathbb{R}^N)} = 1,
\quad \forall n \in \mathbb{N}.

Are these assumptions enough to conclude that $\|u_n\|_{W^{2,2}(\mathbb{R}^N)} \leq C < +\infty$ for some $C > 0$ and all $n \in \mathbb{N}$?

Apart $\Delta u_n$, the norm $\|u_n\|_{W^{2,2}(\mathbb{R}^N)}$ contains also the second mixed derivatives, which probably could make $\|u_n\|_{W^{2,2}(\mathbb{R}^N)}$ unbounded.
However, maybe the additional assumption $\|\nabla u_n\|_{L^2(\mathbb{R}^N)} = 1$ prevents this scenario?

Solutions Collecting From Web of "Boundedness of $L^2$-norms of mixed derivatives of functions from $W^{2,2}(\mathbb{R}^N)$"

Take any $u \in C_c^\infty(\mathbb{R}^N)$ and integrate by parts twice:
$$ \int |\Delta u|^2 = \int \sum_{i,j} u_{ii} u_{jj} = – \int \sum_{i,j} u_{iij} u_j = \int \sum_{i,j} u_{ij} u_{ij} = \int |Du|^2. $$
That is, $\|\Delta u\|_{L^2(\mathbb{R}^N)} = \|D^2 u\|_{L^2(\mathbb{R}^N)}$. By density of $C_c^\infty(\mathbb{R}^N)$ in $W^{2,2}(\mathbb{R}^N)$, the same equality holds for any $u \in W^{2,2}(\mathbb{R}^N)$.

We can apply Sobolev-Poincare inequality $\|u\|_{L^p(\mathbb{R}^N)} \le C_N \| Du \|_{L^2(\mathbb{R}^N)}$ with $p = \frac{2N}{N-2}$ to bound the $L^p$-norm, but the $L^2$-norm seems out of reach.

Edit. Previous version of this answer referred to Poincare’s inequality in $\mathbb{R}^n$ with $p=2$, which is false. It can be seen by considering the family of functions $u_\lambda(x) := u(\lambda x)$ with different values of $\lambda \in \mathbb{R}$ that the only possible value for $p$ is $p = \frac{2N}{N-2}$.