The omitted-value conjecture for Herman rings of meromorphic functions

Let $f$ be a general meromorphic function. An omitted point is a point $z_0\in\mathbb{C}$ such that $f(z)\neq z_0$ for every $z\in\mathbb{C}$. A Herman ring is a periodic doubly connected Fatou component on which an iterate of $f$ is conformally conjugate to an irrational rotation. Omitted-value conjecture. If $f$ omits at least one point in the plane, then $f$ does not have any Herman ring. This would extend known non-existence results for meromorphic functions with omitted values, including cases with multiple poles, a single pole, or a pole that is itself an omitted value; the general assertion remains open.