Positivity on movable curves implies bigness

Let $A$ be a projective manifold, and let ${\mathcal L} \subset \bigwedge^r \Omega^j_A$ be a line bundle. A curve $C$ on $A$ is movable if its deformations fill up $A$. The positivity conjecture. If

$${\mathcal L} \cdot C > 0$$

for every movable curve $C$, then ${\mathcal L}$ is big; in particular, $A$ is of general type. The claim would provide the missing implication from positivity on movable curves to bigness in the setting of differential forms, which is needed in the surrounding argument; the source gives no resolution.