The one-component characterization for bounded symbols of truncated Toeplitz operators

Let $\theta$ be an inner function, and let $K_\theta$ be its model space. Let $\mathcal C_p(\theta)$ denote the class of finite complex Borel measures on the unit circle whose total variation belongs to the corresponding embedding class. The equivalent conditions of Theorem 3 include: every bounded truncated Toeplitz operator on $K_\theta$ admits a bounded symbol, the equality $\mathcal C_1(\theta^2)=\mathcal C_2(\theta^2)$, and the stated factorization property for functions in $H^1\cap\overline z\theta^2H^1_-$. An inner function is one-component when its associated level-set geometry has one connected component.

One-component characterization conjecture. The equivalent conditions of Theorem 3 are fulfilled if and only if $\theta$ is one-component.

The paper proves that one-component inner functions satisfy the equivalent conditions, but states that it does not know whether the converse holds. Thus the proposed characterization remains open.