Model functions with nearly prescribed modulus

Let $\Theta$ be an inner function on the upper half-plane, and let $K_\Theta=H^2\ominus\Theta H^2$ be the corresponding model subspace. A nonnegative measurable function $\omega$ is said to be strongly admissible for $K_{\Theta}$ if there exists a nonzero function $f\in K_{\Theta}$ with $|f|\asymp\omega$. Certain condition sufficient for strong admissibility are given in the case where $\Theta$ is meromorphic.