Peternell's generic positivity conjecture for tangent bundles

Let $X$ be a projective manifold with nef anticanonical bundle $-K_X$. For ample divisors $H_1,\ldots,H_{n-1}$, define generic $(H_1,\ldots,H_{n-1})$-semipositivity of $T_X$ by requiring nonnegative slope for every graded piece of its Harder–Narasimhan filtration with respect to these divisors. Peternell's conjecture. The tangent bundle $T_X$ is generically $(H_1,\ldots,H_{n-1})$-semipositive for every $(n-1)$-tuple of ample divisors $H_1,\ldots,H_{n-1}$. This is stated as a known conjecture about generic positivity under nefness of the anticanonical bundle; no resolution is given in the supplied text.