Asymptotic almost periodicity conjecture for the resolvent norm function

Let $\alpha(\omega)$ denote the norm of $W(-\nu_{0}+i\omega)$, where $W$ is the operator-valued function defined in the preceding construction. Asymptotic almost periodicity conjecture. The function $\alpha(\omega)$ is asymptotically almost periodic in the sense of Bohr as $|\omega|\to\infty$. This conjecture would describe the high-frequency behavior of the resolvent norm and is intended to support uniform approximation arguments on the whole real line; the supplied text gives no resolution of it.