Finite-order automorphism conjecture for surfaces with vanishing irregularity

Let $S$ be a surface with irregularity $q=h^{0,1}=0$, and let $f:S\to S$ be an automorphism of finite order acting trivially on

$$H^0(S,\Omega_S^2).$$

Finite-order automorphism conjecture. The induced map

$$f_*:\mathrm{CH}_0(S)\to\mathrm{CH}_0(S)$$

acts as the identity. This is presented as a special case of the generalized Bloch conjecture, motivated by the study of automorphisms of surfaces; the source does not give a resolution status, so the assertion remains open in this record.