Generalized Yang's conjecture on periodic differential polynomials

Let $f(z)$ be a transcendental entire function and let $n,k$ be positive integers.

Generalized Yang's conjecture. If $f(z)^n f^{(k)}(z)$ is a periodic function, then $f(z)$ is also a periodic function.

This conjecture concerns when periodicity of a differential polynomial forces periodicity of the underlying transcendental entire function. The supplied source does not indicate whether the conjecture has been resolved.