Huybrechts's symplectic-action conjecture for zero-cycles on K3 surfaces

Let $X$ be a $K3$ surface, and let ${\operatorname{Aut}}_s(X)$ be the group of symplectic automorphisms of $X$, meaning automorphisms that act trivially on a nonzero holomorphic $2$-form. Huybrechts's conjecture. The group ${\operatorname{Aut}}_s(X)$ acts trivially on the classical Chow group:

$${\operatorname{Aut}}_s(X)\text{ acts trivially on }{\operatorname{CH}}^2(X).$$

The conjecture concerns the action of symplectic automorphisms on zero-cycles of a $K3$ surface. It was proved in cases where ${\operatorname{Aut}}_s(X)$ is generated by elements of finite order.