Gu's conjecture on reducing subspaces of truncated shifts with one-dimensional defects

Let $B(z)=z^N$. For an inner function $\theta$, let $A_B^\theta$ denote the corresponding truncated Toeplitz operator, and let $\phi_\alpha$ be the disk automorphism used in the definition of $\psi_{\alpha,J}$. Fix a primitive $N$th root of unity $\omega$, and for $J\subset\{0,\dots,N-1\}$ define

$$\psi_{\alpha,J}(z)=\prod_{i\in J}\phi_{\omega^i\alpha}(z).$$

Gu's conjecture. The following are equivalent: $A_B^\theta$ has a reducing subspace such that the restriction has one-dimensional defects; and $\theta(z)=b(z)u(z^N)$ for some inner function $u$, where either $b\equiv1$ or

$$b(z)=\prod_{i=1}^{l}\psi_{\alpha_i,J_i}(z),$$

with $l\le N-1$ and $J_i\subset\{0,\dots,N-1\}$.

The paper presents this as the conjecture left open in Gu's work and aims to prove it, so the equivalence is a resolution of that earlier open problem.