Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries.
Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be Lipschitz injective maps with $\det(df_n)>0$, and suppose that $f_n$ converges to a $C^1$ function $f: \bar\Omega_1 \to \bar\Omega_2$ weakly in $W^{1,2}$, and that $\det(df)>0$ everywhere on $\bar\Omega_1$.
Is it true that $|f^{-1}(y)| \le 1$ a.e. on $\Omega_2$?
Does the answer change if we assume in addition that $f_n|_{K} \to f|_{K}$ strongly in $W^{1,2}$ for every $K \subset \subset \Omega_1$?