Pseudo-effectivity of the second Chern class for strictly nef terminal threefolds

Let $X$ be a terminal projective threefold whose anti-canonical divisor $-K_X$ is strictly nef, meaning that it has positive intersection with every curve. The second Chern class $c_2(X)$ is pseudo-effective, meaning that it lies in the closure of the cone of effective $1$-cycles.

Pseudo-effectivity conjecture. The class $c_2(X)$ is pseudo-effective.

The statement concerns the positivity of the second Chern class on terminal threefolds with strictly nef anti-canonical divisor. It is presented as a conjectural motivation for the paper; the supplied text gives no resolution status.