The conjecture on Fujiki manifolds with holomorphic Cartan geometries

A Fujiki manifold is a compact connected complex space whose reduction is the meromorphic, equivalently holomorphic, image of a compact Kähler manifold. A holomorphic Cartan geometry is a holomorphic Cartan geometry on such a manifold.

Fujiki-manifold conjecture. If a Fujiki manifold admits a holomorphic Cartan geometry, then it admits a flat holomorphic Cartan geometry with the same model, and the Fujiki manifold is Kähler.

This conjecture predicts simultaneous flatness and Kählerness for Fujiki manifolds carrying holomorphic Cartan geometries. The source presents it as an open conjecture.