The $F^{2}$-vanishing form of Beilinson's conjecture

Let $X$ be a smooth projective variety over a number field $k$, let $p$ be a positive integer, and let $F^{\bullet}CH^{p}(X_{\mathbb{C}})_{\mathbb{Q}}$ be the Bloch–Beilinson filtration. Let $F^{2}CH^{p}(X)_{\mathbb{Q}}$ denote the filtration induced from $F^{2}CH^{p}(X_{\mathbb{C}})_{\mathbb{Q}}$ under the natural map $CH^{p}(X)\to CH^{p}(X_{\mathbb{C}})$. Beilinson's $F^{2}$-vanishing conjecture.

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

This is presented as an alternative formulation of Beilinson's conjecture and is therefore not a separate mathematical claim; it asserts that the number-field part of the second Bloch–Beilinson filtration step vanishes.