Boundary Pre-Schwarzian extremal conjecture for the bounded integral means spectrum

Let $B_b(\tau)$ be the universal integral means spectrum for bounded univalent functions, let $N_f$ denote the Pre-Schwarzian derivative of an extremal function $f$, and let $\mathcal{T}$ be the relevant Teichmüller space. Boundary Pre-Schwarzian extremal conjecture. For every $\tau\in\mathbb{C}$, $B_b(\tau)$ has at least one extremal function whose Pre-Schwarzian derivative lies in the boundary of $\mathcal{T}$. The conjecture proposes a boundary characterization of extremals for every complex exponent.