The generically nef tangent bundle conjecture for nef anticanonical manifolds

Let $X$ be a projective manifold, with tangent bundle $T_X$. A vector bundle is generically nef if its restriction to a general complete-intersection curve associated with a polarization is nef; it is sufficiently nef if through every point there is a covering family of curves on whose general member the bundle is nef. A projective manifold is rationally connected if any two general points can be joined by a rational curve.

Generically nef tangent bundle conjecture.

1. If $-K_X$ is nef, then $T_X$ is generically nef.
2. If $-K_X$ is nef, then $T_X$ is sufficiently nef.
3. If $X$ is rationally connected, then $T_X$ is generically ample, and hence sufficiently ample.

These assertions propose positivity properties of tangent bundles in two central classes of projective manifolds: those with nef anticanonical divisor and those that are rationally connected. The source presents them as conjectural, without giving a resolution.