$\newcommand{\SO}[1]{\text{SO}(#1)}$ $\newcommand{\dist}{\operatorname{dist}}$

Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth.

Set $$X=\text{GL}^+_n \cup \{ A \in M_n \, | \text{ the singular values of } \, A \text{ are distinct }\}$$ Here $M_n$ is the space of real $n \times n$ matrices.

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) $?

Can we at least perturb $f$ to make the points where the are recurring singular values isolated? We need to understand what happens to the zeroes of the discriminant of the characteristic polynomial of $df^Tdf$ under perturbation.