Hellerstein–Sheil-Small–Wiman 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–Wiman conjecture. The function $f$ 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. It was stated in the cited earlier work and is presented here as an open conjecture.