A conjecture about meromorphic functions in the plane

Let f : C → C be a non-constant meromorphic function. How many disjoint simply connected regions Gk ⊂ C can exist, such that each preimage f−1(Gk) is connected ? If f has a limit f(∞), that is f is rational, the number of such regions can be at most 2. Indeed, extend f to C, and let Dk = f −1(Gk) ⊂ C be connected. Let fk be the restrictions of f on Dk. Then fk are ramified coverings of degree d = deg f . If some Gk is the sphere, then evidently k = 1. Otherwise, by the Riemann–Hurwitz relation,