Characterization of images of the real line under reduced quasiconformal maps

Let $f\colon \mathbb{C}\to\mathbb{C}$ be a reduced quasiconformal map, and consider the image $f(\mathbb{R})$. A quasisymmetric graph is the image of $\mathbb{R}$ under a suitable graph parametrization, and a vertical line is a line parallel to the imaginary axis. Characterization conjecture. The images of $\mathbb{R}$ under reduced quasiconformal maps $\mathbb{C}\to\mathbb{C}$ are precisely the quasisymmetric graphs and vertical lines. The preceding proposition establishes the converse direction: every nontrivial reduced quasiconformal image is a quasisymmetric graph with quantitative control, and the conjecture asks for the full characterization.