Let $X$ be a projective variety of dimension $n$, let $D$ be a nef divisor on $X$ and let $H$ be an ample divisor. Does $$D \cdot H^{n-1} > 0$$ necessarily hold?
The context I encountered this is the paper Fibre space structures of a projective manifold by Matsushita, where $D$ is the pull-back of a very ample divisor along a morphism $X \to B$ of projective varieties.