Beauville–Voisin conjecture for birational automorphisms of hyper-Kähler varieties

Let $Y$ be a smooth projective hyper-Kähler variety of dimension $2n$. Define

$${\rm CH}_{0}(Y)_{2s}:=\mathrm{S}_{s}{\rm CH}_0(Y)/\mathrm{S}_{s-1}{\rm CH}_0(Y),$$

where $\mathrm{S}_{s}{\rm CH}_0(Y)$ is the Beauville–Voisin filtration piece generated by points whose rational-equivalence orbit has dimension at least $n-s$. For a birational automorphism $\phi\in\mathrm{Bir}(Y)$, let $\phi_{2s}^{{\rm CH}}$ be the induced map on this graded piece. Beauville–Voisin conjecture. If $\phi$ is a (anti)-symplectic birational automorphism, then $\phi_{2s}^{{\rm CH}}$ is the identity in the symplectic case and $(-1)^s\id$ in the anti-symplectic case. This is presented as a trackable consequence of the conjectural Beauville–Voisin splitting and remains open in general.