We know that every proper analytic map from the unit disk to itself is a finite Blaschke product, i.e. $$f(z) = e^{i\theta}\prod_{i=1}^n \dfrac{z - a_i}{1 - \bar{a_i}z}$$ for some $\theta$ and some $a_i \in \Delta$, where the $a_i$ are the zeros.
Is it true that for all $n$, there exist $A, B \in \text{Aut}(\Delta)$ such that $$f(z) = A(p(B(z)))$$ where $p(z) = z^n$? If not, for what $n$ does this hold?