Hard Lefschetz surjectivity conjecture for zero-cycles

Let $X$ be a smooth projective variety of dimension $n$, and let $L$ be an ample line bundle. Let $N^1H^i(X,\mathbb{Q})$ denote the first coniveau piece of cohomology. Hard Lefschetz surjectivity conjecture for zero-cycles. If

$$H^i(X,\mathbb{Q})=N^1H^i(X,\mathbb{Q})\quad\text{for }n\le i<n+r,$$

then intersection induces surjective maps

$$\cdot L^r\colon A^{n-r}(X)_{\mathbb{Q}}\to A^n(X)_{\mathbb{Q}}.$$

The conjecture concerns the expected influence of coniveau on Chow groups and is proposed as a surjectivity counterpart to hard Lefschetz; the source gives no resolution.