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 $\vert f\vert\asymp \omega$. Certain conditions sufficient for strong admissibility are given in the case where $\Theta$ is meromorphic.