Critical-point Schwarzian boundary conjecture for univalent rational functions

Let $\mathcal{U}_R$ be the indicated class of univalent rational functions, let $\mathbb{T}$ be the unit circle, and let $S_R$ denote the Schwarzian derivative of $R$. Critical-point Schwarzian boundary conjecture. If $R\in\mathcal{U}_R$ has at least one critical point in $\mathbb{T}$, then

$$S_R\in\partial\mathbf{S}_Q.$$

The conjecture is motivated by examples with critical points on $\mathbb{T}$ whose Schwarzians are known to lie on this boundary, but the source explicitly says that no proof is known.