Convex shear-class equality conjecture for harmonic mappings

Let $\mathcal{C}^0_H$ be the class of functions in $\mathcal{S}^0_H$ whose image of the unit disk is convex, and let $\mathcal{C}$ be the analytic class of normalized univalent functions mapping the unit disk onto a convex domain. Convex shear-class equality conjecture. For every $f=h+\overline{g}\in\mathcal{C}^0_H$, there exists a real parameter $\theta$ such that $h+e^{i\theta}g$ is univalent and maps the unit disk onto a convex domain; equivalently,

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

This is the convex analogue of the preceding shear-class question. The source provides no evidence that it has been resolved.