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.