The numerical-dimension conjecture for pseudo-effective divisors orthogonal to ample-normal curves

Let $X$ be a projective manifold, let $C\subset X$ be a curve with ample normal sheaf, and let $L$ be a pseudo-effective line bundle. The numerical dimension $\nu(L)$ measures the numerical positivity of $L$.

Numerical-dimension conjecture. If

$$L\cdot C=0,$$

then

$$\nu(L)=0.$$

This would describe the only expected obstruction to a curve with ample normal bundle lying in the interior of the movable cone. The paper proves that under the same orthogonality hypothesis one has $\kappa(L)\leq 0$, and relates the conjecture to the corresponding statement for divisors nef in codimension one; the full assertion remains open.