Ponnusamy–Sairam conjecture on analytic rotations of harmonic mappings

Let $\mathcal S$ be the class of normalized univalent analytic functions, and let $\mathcal S_H^0$ be the class of normalized univalent, sense-preserving harmonic mappings $f=h+\overline g$ in $\mathbb D$ with $g'(0)=0$. For $\theta\in\mathbb R$, define

$$\Phi_\theta=h+e^{i\theta}g$$

and

$$\mathcal S_H^0(\mathcal S)=\{h+\overline g\in\mathcal S_H^0:\Phi_\theta\in\mathcal S\text{ for some }\theta\in\mathbb R\}.$$

Ponnusamy–Sairam's conjecture. One has

$$\mathcal S_H^0(\mathcal S)=\mathcal S_H^0.$$

Equivalently, every $f=h+\overline g\in\mathcal S_H^0$ admits some $\theta\in\mathbb R$ such that $h+e^{i\theta}g\in\mathcal S$. The paper presents this as a new conjecture proposed in earlier work and gives no resolution.