Projective manifolds whose tangent bundle contains a strictly nef subsheaf

Suppose that $X$ is a projective manifold whose tangent bundle $T_X$ contains a locally free strictly nef subsheaf. We prove that $X$ is isomorphic to a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group ${\pi }_1(X)$ is virtually abelian, then $X$ is isomorphic to a projective space.