Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth. Suppose that $df$ is invertible outside a set of Hausdorff dimension $\le n-1$, and that $\text{rank}(df) \ge n-1 $ on $\partial \mathbb{D}^n$.

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 $\text{rank}(df_n) \ge n-1 $ everywhere on $ \mathbb{D}^n $?

Edit:

I am also ready to assume that $\text{rank}(df) \ge n-1 $ outside a set of Hausdorff dimension $\le n-2$.

In fact, I don't know what is the answer even when the set where $\text{rank}(df) \ge n-1 $ is finite, so we can assume that for a start.