Bloch's conjecture for strict Calabi–Yau manifolds

Let $X$ be a strict Calabi–Yau manifold of dimension $n$, and let $\omega_X\in H^0(X,K_X)$ be a nowhere-zero top-degree holomorphic form. Let $Z\in CH^n(X\times X)_{\mathbb Q}$ be a self-correspondence such that $[Z]^*\omega_X=0$. Bloch's conjecture for strict Calabi–Yau manifolds. For every $z\in CH_0(X)_{\mathbb Q,hom}$, one has $Z_*z=0$. This is the strict Calabi–Yau specialization of the generalized Bloch conjecture and is open in general.