Boundary component criterion for Hartogs phenomenon when b > 1

Ascertain whether, for an object X in the categories B of Definition 3.1 that is (b,σ)-compactified by an open immersion X → X' with b > 1, the Hartogs extension property for the structure sheaf O_X implies the existence of a connected component E_i of the boundary Z = X \ X' such that Γ(E_i, O_{X'}|_{E_i}) = C.

Background

Proposition 4.1 provides a sufficient condition for the Hartogs phenomenon in the case b > 1: if some boundary component E_i of Z = X \ X' satisfies Γ(E_i, i^{-1}O_{X'}|_{E_i}) = C, then O_X admits the Hartogs extension property.

The authors raise the converse question—whether the Hartogs phenomenon necessitates the existence of such a boundary component—highlighting an unresolved necessity direction in the characterization of Hartogs extension for (b,σ)-compactified pairs with multiple ends.