Beauville's conjecture on the Chow decomposition of abelian varieties

Let $A$ be an abelian variety over $\mathbb C$ of dimension $g$. Its Beauville decomposition is

$$CH^*(A)_{\mathbb Q}=\bigoplus_{s,i}CH^i_{(s)}(A)_{\mathbb Q},$$

where

$$CH^i_{(s)}(A)_{\mathbb Q}:=\{\alpha\in CH^i(A)_{\mathbb Q}\mid n^*\alpha=n^{2i-s}\alpha\ \forall n\in\mathbb Z\}.$$

Here $n^*$ is induced by multiplication by $n$ on $A$. Beauville's conjecture. $CH^i_{(s)}(A)_{\mathbb Q}$ vanishes for $s<0$, and the cycle class map

$$CH^i_{(0)}(A)_{\mathbb Q}\longrightarrow H^{2i}(A,\mathbb Q(i))$$

is injective for all $i$. This is presented as an outstanding conjecture about cycles on abelian varieties and is used to motivate a higher-dimensional generalization of the paper's vanishing result; apart from the stated context, it remains open in general.