Removal of the auxiliary boundedness assumption in the compactness theorem

Let $v$ be a weight, let $\phi\colon\mathbb D\to\mathbb D$ be an analytic symbol, and let $\psi$ be an analytic function. The boundedness-assumption conjecture. The assumption in Theorem 1 that at least one of the following holds can be dropped:

1. $C_\phi\in\mathcal L(\operatorname{BMOA}_v)$ and either $\operatorname{BMOA}_v\not\subset H^\infty$ or $\psi\in\operatorname{VMOA}_v$;
2. $\left.\psi C_\phi\right|_{\operatorname{VMOA}_v}\in\mathcal L(\operatorname{VMOA}_v)$.

Removing this assumption would give a complete characterization of compactness for weighted composition operators such as $\psi C_\phi$ on weighted BMOA spaces with admissible weights.