The one-component conjecture for bounded truncated Toeplitz operator symbols

Let $I$ be an inner function, let $K_I^2$ denote the corresponding model space, and let a bounded symbol of a truncated Toeplitz operator mean a symbol in $L^\infty(\mathbb{T})$ representing that operator. One-component conjecture. Every bounded truncated Toeplitz operator on $K_I^2$ has a bounded symbol if and only if $I$ is one-component. This is described as a longstanding open conjecture about when bounded truncated Toeplitz operators admit bounded symbols; the paper gives an equivalent formulation in terms of compact operators.