Beilinson extension conjecture for geometric height pairings

Let $X$ be the generic fibre of a regular projective model $\pi:\mathcal{X}\rightarrow B$, with $a_{1}\in CH^{p}_{\operatorname{hom}}(X)_{\mathbb{Q}}$ and $a_{2}\in CH^{q}_{\operatorname{hom}}(X)_{\mathbb{Q}}$, where $p+q=d+1$. An extension of $a_i$ is a cycle $\widetilde{a}_i\in CH^i(\mathcal{X})_{\mathbb{Q}}$ restricting to $a_i$ on $X$.

Beilinson extension conjecture. For each $a_{1}\in CH^{p}_{\operatorname{hom}}(X)_{\mathbb{Q}}$, there exists a Beilinson extension $\widetilde{a}_{1}\in CH^{p}(\mathcal{X})_{\mathbb{Q}}$ such that for every extension $\widetilde{a}_{2}\in CH^{q}(\mathcal{X})_{\mathbb{Q}}$ of $0\in CH^{q}(X)$,

$$\pi_{*}(\widetilde{a}_{1}\cap\widetilde{a}_{2})=0.$$

Such an extension would make the geometric height pairing independent of the choice of extension of the second cycle, even when the model is not smooth. The preceding lemma establishes the analogous vanishing for a smooth model; the conjecture asserts that a suitable extension exists in general.