2 problems

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

Boundary extremal-function conjecture. For each $t\in\mathbb{R}$, $B_b(t)$ has at least one extremal function whose Pre-Schwarzian derivative lies in $\partial\mathcal{T}$.