Generic nefness conjecture for the cotangent and tangent bundles

Let $X$ be a compact Kähler manifold, and let $\alpha_1,\ldots,\alpha_{n-1}$ be Kähler classes. For a coherent sheaf, generically nef means generically nef with respect to the indicated product of Kähler classes. Generic nefness conjecture. (1) If $K_X$ is nef, then $\Omega_X$ is $\alpha_1\cdots\alpha_{n-1}$-generically nef. (2) If $-K_X$ is nef, then $T_X$ is $\alpha_1\cdots\alpha_{n-1}$-generically nef. The conjecture is affirmatively solved in the projective case, but remains open for compact Kähler manifolds; the source attributes the difficulty to the lack of the required foliation theory in that setting.