Exterior-power tangent and cotangent ampleness conjecture

Let $X$ be a projective manifold. For a vector bundle $E$, write $\Lambda^rE$ for its $r$-th exterior power, and let $T_X$ and $T_X^*$ denote the tangent and cotangent bundles of $X$.

Exterior-power ampleness conjecture. The following assertions should hold:

1. If $\Lambda^rT_X$ is strictly nef for some $r>0$, then $K_X^{-1}$ is ample.
2. If $\Lambda^rT_X^*$ is strictly nef for some $r>0$, then $K_X$ is ample.

This proposed general conjecture seeks to characterize ampleness of the canonical or anticanonical bundle from strict nefness of an exterior power of the tangent or cotangent bundle. Its general validity is open.