Co-multiplicative birational Chow–Künneth decomposition conjecture for hyper-Kähler varieties

A birational Chow–Künneth decomposition of a smooth projective variety is co-multiplicative when the induced grading on its zero-cycles is compatible with the diagonal coproduct. Co-multiplicative birational Chow–Künneth decomposition conjecture. Every hyper-Kähler variety admits a co-multiplicative birational Chow–Künneth decomposition. Such a decomposition would provide a coproduct-compatible grading on zero-cycles and is implied by the conjectured existence of a multiplicative Chow–Künneth decomposition; the general assertion remains open.