Ponnusamy–Sheil-Small analytic shear conjecture for normalized harmonic mappings

Let $\mathcal{S}_H^0$ be the subclass of normalized sense-preserving univalent harmonic mappings $f=h+\overline{g}$ with $g'(0)=0$, and let $\mathcal{S}$ be the class of normalized analytic univalent functions. Ponnusamy–Sheil-Small's conjecture. For every function $f=h+\overline{g}\in\mathcal{S}_H^0$, there exists a constant $\theta\in\mathbb{R}$ such that

$$h+e^{i\theta}g\in\mathcal{S}.$$

The paper presents this as a stronger conjecture than the preceding statement and notes that it would also imply an affirmative answer to an open coefficient-estimate question for harmonic mappings in $\mathcal{S}_H^0$.