Mocanu's conjecture on a harmonic mapping class

Let $\mathcal{H}$ be the class of normalized planar harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$, and let $\mathcal{S}_{\mathcal{H}}^0$ denote the standard class of normalized, sense-preserving univalent harmonic mappings with vanishing co-analytic linear coefficient. Define

$$\mathcal{M}=\left\{f=h+\overline{g}\in\mathcal{H}: g'=zh'\ \text{and}\ \operatorname{Re}\left(1+\frac{zh”(z)}{h'(z)}\right)>-\frac{1}{2}\quad(z\in\mathbb{D})\right\}.$$

Mocanu's conjecture. Every mapping in $\mathcal{M}$ belongs to $\mathcal{S}_{\mathcal{H}}^0$; equivalently, $\mathcal{M}\subset\mathcal{S}_{\mathcal{H}}^0$. The conjecture concerns sufficient conditions for univalence and sense preservation of planar harmonic mappings, but the supplied text does not establish its resolution.