Algebraic fibre spaces with strictly nef relative anti-log canonical divisor

Let (X,Delta)$(X,\varDelta)$ be a projective klt pair, and f:X -> Y$f\colon X\rightarrow Y$ a fibration to a smooth projective variety Y$Y$ with strictly nef relative anti-log canonical divisor -(KX/Y+Delta)$-(K_{X/Y}+\varDelta)$. We prove that f$f$ is a locally trivial fibration with rationally connected fibres, and the base Y$Y$ is a canonically polarized hyperbolic manifold. In particular, when Y$Y$ is a single point, we establish that X$X$ is rationally connected. Moreover, when dimX=3$\dim X=3$ and -(KX+Delta)$-(K_X+\varDelta)$ is strictly nef, we prove that -(KX+Delta)$-(K_X+\varDelta)$ is ample, which confirms the singular version of a conjecture by Campana and Peternell for threefolds.