Let $n\geq 3$, $\Omega$, $\Omega'\subset\mathbb{R}^n$ be bounded domains and let $f_m\colon\Omega\to\Omega'$ be a sequence of homeomorphisms with positive Jacobians $J_{f_m}$ and prescribed Dirichlet boundary condition which satisfy the Lusin $(N)$ condition and $\sup_m \int_{\Omega}(|Df_m|^{n-1}+A(|cof Df_m|)+\phi(J_f))<\infty$ (where $A$ and $\phi$ are positive convex functions with superliear growth). Let $f$ be a weak limit of $f_m$ in $W^{1,n-1}$. In this talk we show the proof that $f$ satisfies the $(INV)$ condition of Conti and De Lellis. This talk follows up on the talk of S. Hencl and presents a joint work with S. Hencl and A. Molchanova.