O'Grady's Lagrangian covering family conjecture

Let $M$ be a projective hyperkähler variety. A Lagrangian covering family for $M$ is a closed subscheme $\mathcal{U}\subset M\times B$, pure of dimension $\dim(B)+n$, where $\dim(M)=2n$, such that the general fiber over $B$ is a Lagrangian subvariety of $M$ and the projection of $\mathcal{U}$ to $M$ is all of $M$. O'Grady's conjecture. Any projective hyperkähler variety admits a Lagrangian covering family. This is the same conjecture attributed to O'Grady above; the paper introduces the definition immediately afterward and discusses such families as generalizations of Lagrangian fibrations.