Equality of harmonic and analytic shear classes for normalized univalent mappings

Let $\mathcal{S}^0_H$ be the class of normalized sense-preserving univalent harmonic mappings $f=h+\overline{g}$ in the unit disk with $f_{\overline z}(0)=0$. Let $\mathcal{S}$ be the classical normalized analytic univalent class, and let $\mathcal{S}^0_H(\mathcal{S})$ denote the subclass for which some analytic shear belongs to $\mathcal{S}$. Shear-class equality conjecture. For every $f=h+\overline{g}\in\mathcal{S}^0_H$, there exists at least one $\theta\in\mathbb{R}$ such that $h+e^{i\theta}g\in\mathcal{S}$; equivalently,

$$\mathcal{S}^0_H=\mathcal{S}^0_H(\mathcal{S}).$$

The conjecture asks whether every normalized univalent harmonic mapping admits an analytic univalent shear. The source gives no resolution, so the problem remains open.