Elementary generation in Hardy algebras of the polydisc and ball

Let $d$ be a positive integer. For an open set $U\subseteq\mathbb{C}^d$, let $H^\infty(U)$ be the Banach algebra of bounded holomorphic complex-valued functions on $U$. Let $\mathbb{D}^d$ be the polydisc and let

$$\mathbb{B}_d:=\{(z_1,\ldots,z_d)\in\mathbb{C}^d:|z_1|^2+\cdots+|z_d|^2<1\}$$

be the open unit ball. Let $SL_n(H^\infty(U))$ denote the determinant-one matrices over this algebra, and let $E_n(H^\infty(U))$ be the subgroup generated by elementary matrices.

Hardy-algebra elementary-generation conjecture.

$$SL_n(H^\infty(U))=E_n(H^\infty(U))$$

if $U$ is the polydisc $\mathbb{D}^d$ or the open unit ball $\mathbb{B}_d$.

This is posed as the analogous question for Hardy algebras after proving elementary generation for the disc, ball, and Wiener algebras. The supplied text gives no evidence that this Hardy-algebra claim has been resolved.