The generalized Kummer conjecture for polarized abelian fibrations

Let $X^{2n}$ be an irreducible holomorphic symplectic manifold equipped with an abelian fibration whose fibres have polarization type $(1,\ldots,1,n+1)$. Assume the fibration admits the appropriate multisection. The generalized Kummer conjecture. Then $X$ is birational to a generalized Kummer variety $K_n$. The conjecture is presented as the analogue of the K3 Hilbert-scheme conjecture for generalized Kummer varieties; the source gives no resolution.