Brck conjecture on sharing a finite value with the first derivative

Prove that if f is a non-constant entire function whose hyper-order rho_1(f) is neither a natural number nor infinity, and if f and its first derivative f'(z) share one finite value a counting multiplicities, then there exists a nonzero constant c such that f'(z) - a = c (f(z) - a).

Background

The paper recalls Brck's conjecture as a central open problem in the uniqueness theory of entire functions sharing values with their derivatives. It serves as motivation for the authors' paper of differential-difference analogues and related uniqueness results.

The conjecture posits a strong linear relation between an entire function and its derivative under a value-sharing condition, contingent on the hyper-order of the function. Despite substantial progress, the conjecture remains unresolved in full generality, and the authors reference it to frame their contributions.