Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth.
Question: Do there there exist $f_n \in C^{\infty}(\mathbb{D}^n, \mathbb{R}^n)$ such that $f_n \to f$ in $W^{1,2}(\mathbb{D}^n, \mathbb{R}^n)$ with the property that for every $p \in \mathbb{D}^n $, the singular values of $(df_n)_p$ are all distinct?
I am fine with the $f_n$ being $C^1$, if that matters.