Effective divisors conjecture for holomorphic symplectic fourfolds

Let $(F,g)$ be a polarized irreducible holomorphic symplectic fourfold deformation equivalent to the Hilbert scheme of length-two subschemes of a K3 surface. Let $\operatorname{Pic}_+(F,g)=\{v\in\operatorname{Pic}(F):(v,g)>0\}$, where the pairing is the Beauville form. Effective divisors conjecture. The monoid of effective divisors is generated by the elements of $\operatorname{Pic}_+(F,g)$ with square at least $-2$. This is intended as the divisor-side analogue of the effective-curves description and is stated in comparison with the corresponding description for K3 surfaces. The source gives no resolution.