Sufficient nefness conjecture for cotangent bundles of non-uniruled manifolds

Let $X$ be a non-uniruled projective manifold. A vector bundle $E$ on $X$ is sufficiently nef if, for every $x \in X$, there is a family of curves through $x$ covering $X$ such that $E$ restricted to a general member of the family is nef. Cotangent sufficient-nefness conjecture. The cotangent bundle $\Omega^1_X$ is sufficiently nef. If true, this would imply the conjecture that a non-zero section of $\bigwedge^q T_X \otimes L$, with $L$ numerically trivial and $1\leq q\leq \dim X-1$, cannot have a zero on a non-uniruled manifold.