$C^0$-limit of volume-preserving maps on $\mathbb R^n$

Question

Let $f_k:B_1\rightarrow \mathbb R^n$ be a sequence of injective differentiable volume-preserving maps (i.e. $\mu(f_k(A))=\mu(A)$ for any measurable $A\subset B_1$) that converges uniformly to $f:B_1\rightarrow\mathbb R^n$, where $B_1\subset \mathbb R^n$ is the open unit ball, $\mu$ is the Lebesgue measure. Then can we show that $f$ is also volume-preserving? I can prove that $$\mu(f(A))\ge \mu(A)$$ for any measurable set $A$, but the reverse direction is difficult for me.
I ask this question because in p60 of the book "Symplectic Invariants and Hamiltonian Dynamics (2nd edition)" by Hofer and Zehnder, this statement seems to be implicitly used (although the theorem and idea of argument used in that book are correct), and I suspect it, to be honest.
Current progress: if we are given that $f$ is injective, then using a result in algebraic topology (see Moving of sphere embedding and its interior defined by Jordan-Brouwer separation theorem), we can show that $f_k^{-1}$ converges locally uniformly to $f^{-1}$ on $f(B)$. Then the reverse direction also follows.