Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let $X$ be a compact connected K"ahler manifold such that the holomorphic tangent bundle $TX$ is numerically effective. A theorem of Demailly-Peternell-Schneider says that there is a finite unramified Galois covering $M o X$, a complex torus $T$, and a holomorphic surjective submersion $f: M o T$, such that the fibers of $f$ are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of $f$ are rational and homogeneous. Assume that $X$ admits a holomorphic Cartan geometry. We prove that the fibers of $f$ are rational homogeneous varieties. We also prove that the holomorphic principal ${mathcal G}$--bundle over $T$ given by $f$, where $mathcal G$ is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.