The equality conjecture for the classes of univalent harmonic mappings

Let $\mathcal{S}^0_H$ be the class of normalized univalent sense-preserving harmonic mappings $f=h+\overline{g}$, and let $\mathcal{S}$ denote the class of normalized univalent analytic functions. Define

$$\mathcal{S}^0_H(\mathcal{S})=\left\{h+\overline{g}\in\mathcal{S}^0_H:\ h+e^{i\theta}g\in\mathcal{S}\text{ for some }\theta\in\mathbb{R}\right\}.$$

Equality conjecture.

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

That is, for every function $f=h+\overline{g}\in\mathcal{S}^0_H$, there exists at least one $\theta\in\mathbb{R}$ such that $h+e^{i\theta}g\in\mathcal{S}$. The conjecture asks whether every normalized univalent sense-preserving harmonic mapping admits an analytic univalent shear of this form; it remains open.