Boundedness of composition operators induced by analytic polynomial symbols

Let $v$ be an admissible weight, and let $\phi\colon\mathbb D\to\mathbb D$ be an analytic polynomial symbol. The polynomial-symbol conjecture. The composition operator

$$C_\phi f=f\circ\phi$$

is bounded on $\operatorname{BMOA}_v$.

The paper notes that examples of polynomial symbols with this boundedness property are known. The conjecture asserts the result for every analytic polynomial symbol mapping the disk into itself.