Product decomposition conjecture for characteristic functions of multiplicity-free cnu contractive associators

Let $B_n^*(\mathbb D)$ denote the class of contractive associators of multiplicity $n$ on the unit disc, and call an associator cnu when it is completely non-unitary. A contractive associator is generic when it belongs to one of the classes $B_m^*(\mathbb D)$ with $1\leq m\leq n$ and has the generic property used in the paper. For an associator, let its characteristic function be the associated operator-valued function on $\mathbb D$.

Product decomposition conjecture. The characteristic function of any multiplicity-free cnu contractive associator in $B_n^*(\mathbb D)$ is the pointwise product of the characteristic functions of finitely many generic multiplicity-free contractive associators from

$$\bigcup_{1\leq m\leq n} B_m^*(\mathbb D).$$

This conjecture is prompted by the preceding theorem, which identifies characteristic functions for a family of model associators. It proposes that every multiplicity-free completely non-unitary contractive associator admits a finite factorization into characteristic functions of generic associators; the supplied text gives no resolution status.