Huybrechts' projectivity and bigness conjecture for holomorphic symplectic manifolds

Let $F$ be an irreducible holomorphic symplectic manifold. Let $\operatorname{Pic}(F)$ be its Picard group, equipped with the Beauville form, and suppose that $F$ has a polarization $g$. Define

$$\Lambda_+(F,g)=\{v\in \operatorname{Pic}(F): (v,g)>0,\ (v,v)>0\}.$$

Huybrechts' projectivity and bigness conjecture.

1. $F$ is projective if and only if there exists a class $g\in \operatorname{Pic}(F)$ with $(g,g)>0$.
2. If $g$ is a polarization for $F$, then every class $\lambda\in\Lambda_+(F,g)$ is big.

These statements were conjectured by Huybrechts and are used to connect the Beauville-form positivity conditions with projectivity and birational geometry. The source notes that they had appeared elsewhere with an incomplete proof, but supplies no definitive resolution here.