Univalent analytic shear conjecture for sense-preserving harmonic mappings

Let $f=h+\overline{g}$ be a sense-preserving and univalent harmonic mapping in a simply connected domain $D\subset\mathbb{C}$. Univalent analytic shear conjecture. There exists a constant $\varepsilon\in\overline{\mathbb{D}}$ such that $h+\varepsilon g$ is univalent in $D$.

This conjecture concerns whether every sense-preserving univalent harmonic mapping admits a univalent analytic combination of its analytic and co-analytic parts. It suffices to consider the case $D=\mathbb{D}$; the paper states that the conjecture is weaker than the stronger conjecture proposed by Ponnusamy and Sheil-Small.