Beilinson extension conjecture for vanishing étale cycle classes

Let $X$ be the generic fibre of a regular projective model $\pi:\mathcal{X}\rightarrow B$, and let $CH^{p}_{\operatorname{hom}}(X)_{\mathbb{Q}}$ denote the homologically trivial codimension-$p$ cycles on $X$ with rational coefficients. For an extension $z^{p}_{B}$ of $\alpha^{p}\in CH^{p}_{\operatorname{hom}}(X)_{\mathbb{Q}}$, write $\operatorname{cl}(z^{p}_{B})$ for its étale cycle class.

Beilinson extension conjecture. For every $\alpha^{p}\in CH^{p}_{\operatorname{hom}}(X)_{\mathbb{Q}}$, there exists an extension $z^{p}_{B}$ on $\mathcal{X}$ such that the image of its cycle class in

$$H^{0}_{\operatorname{\acute{e}t}}\left(B,\operatorname{\mathbf{R}}^{2p}\pi_{*}\mathbb{Q}_{\ell}(p)\right)$$

is zero.

The theorem immediately preceding this statement shows that such a vanishing forces the associated boundary class to vanish and hence removes the spare term in the comparison of height pairings. The conjecture asserts that an extension with this property can always be chosen.