Bloch's conjecture for symplectic automorphisms of K3 surfaces

Let $X$ be a smooth projective K3 surface, and let ${\rm Aut}^s(X)$ denote its group of symplectic automorphisms. For $\phi\in{\rm Aut}^s(X)$, write

$$\phi^{{\rm CH}}:{\rm CH}_0(X)_{{\rm hom}}\to {\rm CH}_0(X)_{{\rm hom}}$$

for the induced morphism. Bloch's conjecture. For every $\phi\in {\rm Aut}^s(X)$, the induced morphism $\phi^{{\rm CH}}$ is the identity. This has been confirmed when the Néron–Severi lattice satisfies the stated rank conditions, when $\phi$ has finite order, and for elliptic K3 surfaces where $\phi$ preserves the elliptic fibration; the general case remains open.