Green–Griffiths implication for the second Chow-group filtration over number fields

Let $k$ be a number field and let $X$ be a smooth projective variety over $k$. Write $X_{\mathbb C}=X\times_{\operatorname{Spec}(k)}\operatorname{Spec}(\mathbb C)$. The filtration on $CH^p(X)_{\mathbb Q}$ is induced from the second step of the filtration on $CH^p(X_{\mathbb C})_{\mathbb Q}$ under the natural map

$$CH^p(X)\longrightarrow CH^p(X_{\mathbb C}).$$

Green–Griffiths implication. The induced second filtration step vanishes:

$$F^2CH^p(X)_{\mathbb Q}=0.$$

This is presented as an implication proposed by Green and Griffiths in the context of Hodge-theoretic filtrations on Chow groups. Its general validity for smooth projective varieties over number fields remains open in the source.