The interior-of-the-Mori-cone conjecture for curves with ample normal bundle

Let $X$ be a projective manifold and let $C\subset X$ be an irreducible curve. Assume that the normal sheaf $N_{C/X}$ is ample. Write $\overline{NE}(X)$ for the Mori cone of curves on $X$.

Interior-of-the-Mori-cone conjecture. The class $[C]$ lies in the interior of

$$\overline{NE}(X).$$

Equivalently, every nef line bundle $L$ satisfying $L\cdot C=0$ is numerically trivial. The conjecture is motivated by the duality between the effective and nef cones; the paper proves it when $C$ moves in a family covering $X$, while the general case remains open.