The extension conjecture for normalized univalent maps

Let $n\geq2$, let $f\in{\mathcal{S}}$ be normalized and univalent on ${\mathbb{B}^n}\subseteq{\mathbb{C}^n}$. Let $r>1$ and let $F\in{\mathcal{S}}(r{\mathbb{B}^n})$ be a univalent extension satisfying $F|_{{\mathbb{B}^n}}=f$.

Extension conjecture. Then $f\in{\mathcal{S}}^1$, that is, $f$ embeds into a normalized Loewner chain. The source states that this is equivalent to the generalized Andérsen–Lempert conjecture and remains conjectural there.