Existence of l.c.K. manifolds realizing case (3)

Determine whether there exist compact l.c.K. manifolds with nonnegative bisectional curvature whose universal covers are conformally equivalent to the product of an incomplete Kähler manifold and a positive-dimensional complex Euclidean space, as specified in case (3) of the structural trichotomy.

Background

The structural theorem identifies a third possibility for a compact l.c.K. manifold with nonnegative bisectional curvature: its universal cover is conformally equivalent to $(L,g_L)\times(\mathbb{C}^k,g_0)$, where $L$ is incomplete Kähler and $k>0$. The paper proves that this alternative cannot occur when the fundamental group is isomorphic to $\mathbb{Z}$, but does not rule it out in general.

The authors note that Inoue surfaces have the geometric form appearing in case (3), but their tangent bundles are not nef and therefore they cannot admit Hermitian metrics with nonnegative Chern bisectional curvature. The existence of any compact l.c.K. examples satisfying both the curvature condition and case (3) consequently remains unresolved.