Hellerstein–Sheil-Small–Wittich classification conjecture for real meromorphic functions

Let $f$ be a real transcendental meromorphic function in the plane with at least one pole. Assume that all zeros and poles of $f$, $f'$ and $f”$ are real, and that all poles of $f$ are simple.

Hellerstein–Sheil-Small–Wittich conjecture. Then $f$ satisfies

$$f(z)=C\tan(az+b)+Dz+E,\qquad a,b,C,D,E\in\mathbb{R}.$$

This conjecture concerns the classification of real meromorphic functions whose first two derivatives have only real zeros and poles. The source attributes it to Hellerstein, Sheil-Small and Wittich; its resolution is not specified here.