Conjecture on conjugation-invariant matrix representations of truncated Toeplitz operators

Let $M$ be a complex symmetric matrix, and suppose that $M$ is unitarily equivalent to a truncated Toeplitz operator (TTO). Let $B$ be an inner function, let $K_B$ be the corresponding model space, and let $C_B$ denote its canonical conjugation. A basis of $K_B$ is $C_B$-real if it is fixed by $C_B$. Conjugation-invariant representation conjecture. There exist an inner function $B$ and a $C_B$-real basis for $K_B$ such that $M$ is the matrix representation of a TTO on $K_B$ with respect to this basis; equivalently, every unitary equivalence between a complex symmetric matrix and a TTO arises from a $C_B$-real matrix representation. The conjecture concerns whether arbitrary unitary equivalences can be realized through the canonical conjugation associated with a model space; the supplied text gives no resolution, so its status remains open.