The $C_{\theta}$-real matrix representation conjecture for truncated Toeplitz operators

Let $M$ be a complex symmetric matrix. Let an inner function be a bounded analytic function on the unit disk whose radial boundary values have modulus one almost everywhere, and let $K_u=H^2\ominus uH^2$ be the corresponding model space. A basis of $K_u$ is $C_{\theta}$-real when it is fixed by the conjugation $C_{\theta}$. A matrix is unitarily equivalent to a truncated Toeplitz operator (TTO) if it is unitarily similar to an operator of the form $A_\varphi^u=P_{K_u}M_\varphi|_{K_u}$ for some symbol $\varphi$.

The $C_{\theta}$-real matrix representation conjecture. If $M$ is a complex symmetric matrix and is unitarily equivalent to a TTO, then there exist an inner function $u$ and a $C_{\theta}$-real basis for $K_u$ such that $M$ is the matrix representation of a TTO on $K_u$ with respect to this basis. Equivalently, every unitary equivalence between a complex symmetric matrix and a TTO should arise from a $C_{\theta}$-real matrix representation.

The conjecture asks whether the unitary equivalence result for complex symmetric matrices and TTOs can always be realized using a basis compatible with the model-space conjugation. The preceding corollary gives examples showing that a related, more restrictive question about modified Clark bases has a negative answer, but does not resolve this broader question.