Uniqueness conjecture for boundary Schwarzian data in the universal Teichmüller space

Let $\overline{\mathbf{S}}_q$ be the closure of the universal Teichmüller space in the Schwarzian model. For $\phi\in\overline{\mathbf{S}}_q$, obtain $f_\phi$ as the locally convergent limit of normalized univalent functions whose Schwarzians converge to $\phi$.

Uniqueness conjecture. Let $\phi\in\overline{\mathbf{S}}_q$. Then the function $f_\phi$ obtained in this way is unique, so that the IMS functional

$$I_{\overline{T}_2}:\phi\mapsto\beta_{f_\phi}(t),\qquad \phi\in\overline{\mathbf{S}}_q,$$

is well-defined.

The source explains that uniqueness is known in the interior but has not been proved on the boundary; consequently, well-definedness of the extended IMS functional remains open.