Extension criterion on arbitrary open subsets of cl(D^n)

Determine whether the following equivalence holds: for any natural number n and any complex Banach space X, and for any open set U contained in cl(D^n) (the closure of D^n inside the maximal ideal space M(H∞(D^n))), an X-valued holomorphic function f on U ∩ D^n admits a continuous extension to U if and only if f(V) is relatively compact in X for every relatively compact subset V of U.

Background

The paper proves a Runge-type approximation theorem for Banach-valued holomorphic functions on polydisks and derives Corollary 1.2, which characterizes when such functions extend continuously to T_n^{-1}(U) ⊂ cl(D^n) for open U ⊂ M^n: namely, precisely when the image of f is relatively compact on every relatively compact subset of U.

The map T_n: cl(D^n) → M^n is identity on D^n but is not one-to-one outside D^n, and there exist many open subsets of cl(D^n) that are not preimages of open subsets of M^n under T_n. Motivated by this, the authors conjecture that the same extension criterion should hold directly for arbitrary open subsets U ⊂ cl(D^n), not only for sets of the form T_n^{-1}(U). Establishing this would extend Corollary 1.2 beyond the class of preimage-open sets under T_n.