Approximating the inverse matrix of the G-limit through changes of variables in the plane

Let $A_j$ be a sequence of coercive symmetric matrices of $L^\infty(\mathbb{R}^2)^{2\times 2}$ with $det \, A_j=1$ which $G$-converges to $A$. We prove that there exists a sequence of $K$-quasiconformal mappings $F_j$ which converge locally uniformly to a $K$-quasiconformal mapping $F$ such that $A_j^{-1}\circ F_j^{-1}$ $G$-converges to $A^{-1}\circ F^{-1}$. The result is specific to the two dimensional case but a similar result holds in dimension $1$.