Approximation-extension conjecture for Banach-valued holomorphic functions on the polydisk maximal ideal space

Let $\mathbb D^n$ be the unit polydisk, let $X$ be the Banach space in which $f$ takes values, and let $\operatorname{cl}(\mathbb D^n)$ denote the closure of $\mathbb D^n$ in the relevant maximal ideal space. For an open set $\widehat U\subset\operatorname{cl}(\mathbb D^n)$, put

$$U=\widehat U\cap\mathbb D^n.$$

Here $V\Subset\widehat U$ means that $V$ is relatively compact in $\widehat U$.

Extension conjecture. An $X$-valued holomorphic function $f$ on $U$ admits a continuous extension to $\widehat U$ if and only if $f(V)\Subset X$ for every subset $V\Subset\widehat U$ of $U$.

This conjecture asks whether the extension criterion from the preceding corollary remains valid for arbitrary open subsets of the closure of the polydisk, rather than only for inverse images of open subsets under the map to the quotient maximal ideal space. Its resolution is not stated in the source.