Pseudo-effectivity conjecture for the second Chern class of varieties with nef anticanonical divisor

Let $X$ be a variety of dimension $n$ with only terminal singularities, and assume that $-K_X$ is nef. Let $H_1,\ldots,H_{n-2}$ be ample Cartier divisors on $X$.

Pseudo-effectivity conjecture for the second Chern class.

$$c_2(X)\cdot H_1\cdots H_{n-2}\geq 0.$$

This conjecture concerns the pseudo-effectivity of the second Chern class for varieties with nef anticanonical divisor. The source explicitly says that it remains open when $1<\nu(-K_X)<n$.