The embedding conjecture for normalized univalent maps

Let $n\foldgeq2$. Let ${\mathcal{S}}$ be the space of normalized univalent maps on the unit ball ${\mathbb{B}^n}\subseteq{\mathbb{C}^n}$, and let ${\mathcal{S}}^1$ be the subclass consisting of maps that embed into a normalized Loewner chain. Let ${\mathcal{S}}({\mathbb{C}^n})$ denote the space of normalized entire univalent maps.

Embedding conjecture.

$${\mathcal{S}}={\mathcal{S}}^1.$$

Equivalently,

$${\mathcal{S}}={\mathcal{S}}({\mathbb{C}^n})\circ{\mathcal{S}}^0,$$

where ${\mathcal{S}}^0$ is the class of maps embedding into normal Loewner chains. This asks whether every normalized univalent map on the ball in dimension at least two embeds into a normalized Loewner chain; the question is stated as open in the source.