Kawamata's second-Chern-class inequality for nef anticanonical bundles

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

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

The paper presents this as a conjecture and discusses related results in the dual nef-canonical case and in dimension three; the supplied text does not resolve the general assertion.