Uniqueness from Two Nonzero Shared Values with the Derivative

Determine whether a non-constant meromorphic function f on the complex plane that shares two nonzero finite values with its derivative f' (in the sense that f(z)=a if and only if f'(z)=a for each of the two values) must satisfy f ≡ f'.

Background

In the classical shared-value setting for a meromorphic function f and its derivative f', Gundersen and independently Mues–Steinmetz proved that if f and f' share three finite values, then f and f' must be identical (f ≡ f'). This is a cornerstone result within Nevanlinna theory for the special pair (f, f').

The paper notes that it remains unresolved whether the threshold of three shared values can be lowered to two when the two shared values are nonzero. Resolving this would sharpen the classical uniqueness theory for meromorphic functions and their derivatives.