Bloch's conjecture for hyperkähler varieties with non-symplectic automorphisms

Let $X$ be a hyperkähler variety of dimension $2m$, and let $\sigma\in\operatorname{Aut}(X)$ be a non-symplectic automorphism of order $k$ with $k>m$. Then

$$\bigl(\sigma+\sigma^2+\ldots+\sigma^k\bigr)_*:A^{2m}_{hom}(X)\to A^{2m}(X)\text{ is zero}.$$

Bloch's conjecture. The sum of the actions of the powers of $\sigma$ annihilates homologically trivial zero-cycles. This prediction follows in the source from the correspondence formulation of Bloch's conjecture; the supplied status evidence does not report a resolution of this hyperkähler case in general.