Shen–Vial's multiplicative Chow–Künneth conjecture for hyper-Kähler varieties

A hyper-Kähler variety is a smooth projective variety with the usual hyper-Kähler structure; a multiplicative Chow–Künneth decomposition is a Chow–Künneth decomposition compatible with the intersection product. Shen–Vial's conjecture. Any hyper-Kähler variety admits a multiplicative Chow–Künneth decomposition. This conjecture is known for several classes, including K3 surfaces, Hilbert schemes of points on K3 surfaces, generalized Kummer varieties, and all hyper-Kähler varieties birational to one of these examples, but remains open in general.