Voisin's surface decomposability conjecture for hyper-Kähler varieties

A hyper-Kähler variety $X$ of dimension $2n$ is called surface decomposable if there exist a smooth projective variety $\Gamma$, smooth projective surfaces $S_1,\ldots,S_n$, and generically finite morphisms from $\Gamma$ to $X$ and to $S_1\times\cdots\times S_n$ such that every global $2$-form on $X$ pulls back to a sum of pullbacks of global $2$-forms from the surfaces. Voisin's surface decomposability conjecture. Every hyper-Kähler variety is surface decomposable. Voisin established the claim in several cases, including Hilbert schemes of points on K3 surfaces, generalized Kummer varieties, Fano varieties of lines on cubic fourfolds, and LLSvS varieties; the general case remains open.