Kawamata's nonnegativity conjecture for the second Chern class

Let $X$ be a compact Kähler manifold of dimension $n$ with nef anticanonical bundle, and let $\omega_1,\ldots,\omega_{n-2}$ be nef classes. Kawamata's second-Chern-class conjecture. One has

$$\int_X c_2(T_X)\wedge\omega_1\wedge\cdots\wedge\omega_{n-2}\geq 0.$$

This conjecture concerns the positivity of the second Chern class for manifolds with nef anticanonical bundle; the supplied text gives no resolution of the full statement.