Hellerstein–Shen–Williamson conjecture on real meromorphic functions

Let $f$ be a real transcendental meromorphic function in the plane with at least one pole, meaning that $f(\mathbb R)\subseteq\mathbb R\cup\{\infty\}$. Assume that all zeros and poles of $f$, $f'$ and $f”$ are real, and that all poles of $f$ are simple. Hellerstein–Shen–Williamson conjecture. Then

$$f(z)=C\tan(az+b)+Dz+E,$$

where $a,b,C,D,E\in\mathbb R$. The paper proves a special case of this classification problem for real meromorphic functions; the supplied evidence does not establish that this full conjecture has been resolved.