Boundary extremal-function conjecture for the bounded integral mean spectrum

Let $B_b(t)$ be the universal integral mean spectrum for bounded univalent functions, let $f$ be an extremal function when $\beta_f(t)=B_b(t)$, and let $N_f$ denote its Pre-Schwarzian derivative. Let $\mathcal{T}$ be the relevant Teichmüller space of Pre-Schwarzian derivatives.

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 $\partial\mathcal{T}$.

The conjecture generalizes the examples known for $t\geq2$ and for $t\leq t_0$, asserting that an extremizer can always be chosen on the boundary. The source does not give a resolution.