KLP's compression symmetry characterization conjecture for truncated Toeplitz operators

Let $u$ be an inner function on the unit disc, let $K_u=H^2\ominus uH^2$, and let $\mathcal{T}_u$ be the space of truncated Toeplitz operators on $K_u$. For an inner divisor $v$ of $u$, write $K_v\subset K_u$ for the corresponding model space, $P_v$ for the orthogonal projection onto $K_v$, and $C_v$ for its canonical conjugation. KLP's conjecture. A bounded linear operator $A$ on $K_u$ belongs to $\mathcal{T}_u$ if and only if, for every inner divisor $v$ of $u$, the compression $P_vA|_{K_v}$ is $C_v$-symmetric. The conjecture seeks an intrinsic characterization of truncated Toeplitz operators through the symmetries of all compressions to model subspaces; the source gives no resolution status.