Algebraic cycles on certain hyperkaehler fourfolds with an order $3$ non-symplectic automorphism

Let $X$ be a hyperk"ahler variety, and assume $X$ has a non-symplectic automorphism $\sigma$ of order $>\frac{1}{2}\dim X$. Bloch’s conjecture predicts that the quotient $X\mathrm{/}<\sigma >$ should have trivial Chow group of $0$-cycles. We verify this for Fano varieties of lines on certain special cubic fourfolds having an order $3$ non–symplectic automorphism.