Covariant weak Lefschetz conjecture for zero-cycles

Let $X$ be a smooth projective variety, let $A_0X$ be its Chow group of zero-cycles, and let $N^1H^i(X,\mathbb{Q})$ denote the first step of the coniveau filtration. Let $Y\subset X$ be a smooth complete intersection of dimension $\ell$.

Covariant weak Lefschetz conjecture. Suppose

$$H^i(X,\mathbb{Q})=N^1H^i(X,\mathbb{Q})$$

for $i\in[\ell+1,n]$. Then the push-forward map

$$A_0Y_{\mathbb{Q}}\ \longrightarrow\ A_0X_{\mathbb{Q}}$$

is surjective.

This formulation avoids the conjectural filtration on $A_0$. It is motivated by the expectation that the relevant quotient of $A_0X$ is controlled by the cohomology groups in degrees $2n$ through $2n-\ell$.