Schwarzian norm conjecture for maximizers of the functional Lambda

Let $\mathcal{S}$ be the class of normalized univalent functions, let $\overline{\mathbf{S}}$ be the indicated closed Schwarzian space, let $S_f$ be the Schwarzian derivative of $f\in\mathcal{S}$, and let $\Lambda_\alpha$ be the functional in the source. Lambda-maximizer Schwarzian norm conjecture. For every $\alpha>0$, if $\Lambda_\alpha$ attains at least one maximum at $S_f\in\overline{\mathbf{S}}$ for some $f\in\mathcal{S}$, then

$$\|S_f\|_{E_2}=6.$$

The preceding text asks separately whether maxima exist; this conjecture concerns the norm of any such maximizer under the stated attainment hypothesis.