Beilinson's conjecture on Chow classes and the Hochschild–Serre filtration

Let $X$) be a smooth projective variety defined over a number field $k$, and let $CH^n(X)_{\mathbb Q}$ denote its codimension-$n$ Chow group with rational coefficients. For $i=0,1,2$, consider the Hochschild–Serre terms

$$H^i\left(k,H^{2n-i}(X_{\bar k},\mathbb Q_\ell(n))\right).$$

Beilinson's conjecture. With notation as above, if a class in $CH^n(X)_{\mathbb Q}$ vanishes in $H^i\left(k,H^{2n-i}(X_{\bar k},\mathbb Q_\ell(n))\right)$ for $i=0,1$, then it does for $i=2$.

This is a formulation of Beilinson's expected injectivity of the cycle-class map into Deligne cohomology, expressed through the first terms of the Hochschild–Serre filtration on continuous $\ell$-adic cohomology. The source presents it as a conjectural prediction, with no resolution supplied here.