Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth, with $\text{rank}(df) \ge n-1$ everywhere.

Set $X=\text{GL}^+_n \cup \{ A \in M_n \, | \,\,\sigma_1(A) < \sigma_2(A)\},$

where $M_n$ is the space of real $n \times n$ matrices, and $\sigma_1(A) \le \sigma_2(A) \le \dots \sigma_n(A)$ are the singular values of $A$.

Note that $X \subseteq \{ A \in M_n \, | \, \text{rank}(A) \ge n-1 \}$, since if $\sigma_1=0$ we must have $\sigma_2>0$. Writing $X$ as a disjoint union, $$X=\text{GL}^+_n \cup (\text{rank}=n-1) \cup (\text{GL}^-_n \cap \{\sigma_1 < \sigma_2\}).$$

Question: Do there there exist $f_n \in C^{\infty}(\mathbb{D}^n, \mathbb{R}^n)$ such that $f_n \to f$ in $W^{1,2}(\mathbb{D}^n, \mathbb{R}^n)$ and $df_n \in X$ everywhere on $ \text{int}(\mathbb{D}^n) $?