Suppose that $\varphi\colon \mathbb{C} \to \mathbb{C}$ is a quasiconformal map (i.e., with $\mu_\varphi < k < 1$ on $\mathbb{C}$) such that its dilatation $\mu_\varphi$ is supported in some compact set $K \subset \mathbb{C}$. Is it true that there exists a complex-affine map $M(z) = az+ b$ such that $|\varphi(z) - M(z)|$ converges to $0$ as $|z| \to \infty$?
What if the dilatation of $\varphi$ is not supported in a compact set, but in a set which is "very small near infinity"? For instance, what if the dilation of $\varphi$ is supported in $\{x + iy: x \geq 0, -e^{-x} \leq y \leq e^{-x}\}$.