O'Grady's covering-family conjecture for Lagrangian subvarieties

Let $X_n$ be the hyperkähler variety under consideration, let $\Omega_n\subset X_n$ be the distinguished Lagrangian subvariety, and let a family over a base $S$ be a morphism $\pi:\mathcal{Z}_S\to X_n$. Here a Lagrangian subvariety is a subvariety on which the holomorphic symplectic form restricts to zero and whose dimension is half that of $X_n$.

O'Grady's conjecture. There is a positive integer $a\geq 1$ and a family of Lagrangian subvarieties $\pi:\mathcal{Z}_S\to X_n$ over a base $S$ such that $\mathcal{Z}_S$ covers $X_n$, i.e. $\pi$ is surjective; for all $s\in S$, $\mathcal{Z}_s=\sum_i m_i\Gamma_i\subset X_n$ is an effective cycle with each $\Gamma_i\subset X_n$ a Lagrangian subvariety; and there is an $s_0\in S$ such that $\mathcal{Z}_{s_0}=a\Omega_n$.

This conjecture predicts a covering family whose members include a positive multiple of the distinguished Lagrangian subvariety, as part of the proposed construction of new hyperkähler varieties from Lagrangian subvarieties. Its resolution status is not specified in the supplied text.