Hartogs extension conjecture for Kähler holomorphic fibrations

Let $E \to M \to B$ be a holomorphic fibration whose total space is Kähler. A complex manifold is a Hartogs extension target for local biholomorphisms when local biholomorphisms into it extend across the relevant Hartogs sets. The Kähler fibration extension conjecture. If the base $B$ and the fiber $E$ are Hartogs extension targets for local biholomorphisms, then the total space is also a Hartogs extension target for local biholomorphisms. Non-Kähler examples in the surrounding discussion show that the Kähler hypothesis is significant; the claim proposes that it prevents the extension failure exhibited there.