Boundary Schwarzian extremal conjecture for the bounded integral means spectrum

Let $B_b(\tau)$ be the universal integral means spectrum for bounded univalent functions, let $S_f$ denote the Schwarzian derivative of an extremal function $f$, and let $\mathbf{S}_Q$ be the indicated Schwarzian-derivative space. Boundary Schwarzian extremal conjecture. For every $\tau\in\mathbb{C}$, $B_b(\tau)$ has at least one extremal function whose Schwarzian derivative lies in the boundary $\partial\mathbf{S}_Q$ of $\mathbf{S}_Q$. This is stated as a consequence that would follow from the corresponding Pre-Schwarzian conjecture.