Trivial-action conjecture for automorphisms of surfaces

Let $S$ be a smooth projective surface, and let $G\subset\mathrm{Aut}(S)$ be a group of automorphisms. Assume that every element of $G$ acts trivially on $H^{2,0}(S)$. Trivial-action conjecture. Then $G$ acts trivially on

$$\mathrm{CH}_0(S)_{\mathrm{alb}}.$$

This is obtained in the source from the correspondence conjecture by taking the difference between an automorphism's graph and the diagonal. The supplied text does not state whether it has been resolved.