Hartogs phenomenon for locally free sheaves under finite compactly supported cohomology

Determine whether, for a noncompact normal complex analytic variety X with a single topological end, any Hartogs pair (K, X), and any locally free O_X-module F of finite rank, the finiteness of the first compactly supported cohomology group dim_C H^1_c(X, F) implies that F admits the Hartogs extension phenomenon with respect to (K, X).

Background

Proposition 2.1 establishes that for noncompact normal complex analytic varieties with one topological end, finiteness of dim_C H^1_c(X, O_X) yields the Hartogs extension property for the structure sheaf via the Andreotti–Hill trick. However, the authors note that this trick does not apply to arbitrary locally free O_X-modules of finite rank.

They reduce the general problem to holomorphic line bundles associated with effective Cartier divisors and solve it for complex manifolds under additional assumptions (Theorem 2.3), but the general case for arbitrary locally free sheaves on normal varieties remains unresolved.