Nonnegative-curvature Kähler or Vaisman metrics

Determine whether every compact complex manifold that admits an l.c.K. metric with nonnegative Chern bisectional curvature and also admits a Kähler metric, respectively a Vaisman metric, must admit a Kähler metric, respectively a Vaisman metric, with nonnegative Chern bisectional curvature.

Background

The paper proves a trichotomy for compact l.c.K. manifolds with nonnegative Chern bisectional curvature: the underlying complex manifold admits a Kähler metric, admits a Vaisman metric, or has a universal cover conformally equivalent to the product of an incomplete Kähler manifold and a positive-dimensional complex Euclidean space. In the first two alternatives, the proof establishes only the existence of the relevant Kähler or Vaisman metric and does not show that this metric retains nonnegative Chern bisectional curvature.

The unresolved problem asks whether the curvature condition can be preserved when passing from the initially given l.c.K. metric to a Kähler or Vaisman metric on the same compact complex manifold.