Steinness criterion for deformation families of canonical extensions

Determine whether the fibers of the deformation family of the canonical extension W_M of a compact Kähler manifold M are all Stein if and only if the tangent bundle of every fiber of the underlying complex analytic family of compact complex manifolds is numerically effective (nef).

Background

The paper constructs a deformation family of the canonical extension W_M associated with a compact Kähler manifold M from a sufficiently small Kähler deformation family of M. The fiber over the central point is holomorphically isomorphic to W_M, while the other fibers are canonical extensions of the corresponding compact fibers.

The cited conjecture proposes a necessary-and-sufficient criterion for all members of this canonical-extension deformation family to be Stein: the tangent bundle of every compact fiber in the underlying deformation family should be nef. The paper establishes the sufficient direction using an external theorem: if every tangent bundle is nef, then all canonical-extension fibers are Stein. The converse implication remains unresolved in the stated material.