The K3 Hilbert-scheme conjecture for Jacobian abelian fibrations

Let $X^{2n}$ be an irreducible holomorphic symplectic manifold equipped with an abelian fibration whose fibres are isomorphic to Jacobians of curves. A multisection is a subvariety meeting the general fibre in finitely many points; the source assumes the appropriate multisection described in its discussion. The K3 Hilbert-scheme conjecture. If the fibration admits the appropriate multisection, then $X$ is birational to

$$S^{[n]},$$

the Hilbert scheme of $n$ points on a K3 surface $S$. The conjecture is known for $n=2$ by Markushevich, while the source discusses possible issues in higher dimensions, including a possible example for $n=5$; it remains open in general.