Conjecture on the pseudo-effective cone of projective irreducible holomorphic symplectic manifolds

Let $X$ be a projective irreducible holomorphic symplectic (IHS) manifold with Picard number greater than $3$. Let $\operatorname{Neg}(X)$ denote the set of negative divisors used in the source, and let $\overline{\operatorname{Eff}(X)}$ be the closure of the effective cone. Pseudo-effective cone conjecture. Either

$$\operatorname{Neg}(X)=\emptyset$$

and $\overline{\operatorname{Eff}(X)}$ is circular, or

$$\operatorname{Neg}(X)\neq\emptyset$$

and

$$\overline{\operatorname{Eff}(X)}=\overline{\sum_{E\in\operatorname{Neg}(X)}\mathbf{R}^{\geq 0}[E]}.$$

The conjecture is proposed as a reasonable consequence of the results in the paper; related descriptions are established for the four currently known deformation classes, but the general case remains open.