A counterexample to conformal invariance for increasing radial weights

Let $\mathbb D$ denote the unit disk, let $v\colon\mathbb D\to(0,\infty)$ be an increasing radial weight that is not admissible, let $\hat\phi\in\operatorname{Aut}(\mathbb D)$, and let $f\in\operatorname{BMOA}_v$. The non-invariance conjecture. There exist such $v$, $\hat\phi$, and $f$ for which

$$f\circ\hat\phi\notin\operatorname{BMOA}_v.$$

This would exhibit failure of invariance under composition with disk automorphisms for a suitable non-admissible weight; the supplied text presents the claim among open conjectures and discusses why constructing the required function remains difficult.