Conjecture characterizing truncated Toeplitz operators by conjugation symmetries

Let $\theta$ be a nonconstant inner function, and let $A\in L(K^2_\theta)$. For every nonconstant inner function $\alpha$ dividing $\theta$, let $A_\alpha=P_\alpha A_{|K^2_\alpha}$.

Characterization conjecture. $A\in\mathcal{T}(\theta)$ if and only if $A_\alpha$ is $C_\alpha$-symmetric for every nonconstant inner function $\alpha$ dividing $\theta$.

The preceding results establish this characterization in several important cases, including finite Blaschke products and the setting treated in the paper. The conjecture proposes that the same criterion holds for arbitrary nonconstant inner functions.