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Theorem 1.2 in Q. Xie `On Pseudo-Effectivity of the Second Chern Classes for Terminal Threefolds' shows that $c_2(X)$ is at least pseudoeffective (i.e., a limit of effective classes). On the other hand, since $X$ is Fano, the Cone theorem shows that the cone of curves is rational polyhedral, spanned by finitely many classes of rational curves, so $c_2(X)$ is in fact effective.