Non-vanishing conjecture for strictly nef anti-canonical divisors

Let $X$ be a projective rationally connected manifold such that the anti-canonical divisor $-K_X$ is strictly nef. The Kodaira dimension of a divisor is denoted by $\kappa$.

Non-vanishing conjecture for strictly nef anti-canonical divisors. One should have

$$\kappa(-K_X)\geq 0.$$

Together with the weak Campana–Peternell's conjecture, this is stated to be equivalent to the Campana–Peternell conjecture. The paper proves non-vanishing results in several cases, including for rationally connected fourfolds under suitable numerical assumptions, but the general statement remains open.