Equality of the Loewner and univalent-function classes

Let $\mathcal D_n$ denote the unit polydisc, and let $E(\mathcal D_n)$ and $F(\mathcal D_n)$ be the two classes of normalized mappings defined in the paper, with tilded versions allowing the corresponding biholomorphic changes of coordinates. The equality conjecture. For every $n\in\mathbb N$,

$$E(\mathcal D_n)=F(\mathcal D_n),\qquad \widetilde E(\mathcal D_n)=\widetilde F(\mathcal D_n).$$

The source gives density results but does not establish these equalities.