The motivic origin conjecture for Beilinson's geometric height pairing

Let $B$ be the base and $X$ the smooth proper variety over the relevant function field, with $d=\operatorname{dim}X$ and $p+q=d+1$. Write $CH^p_{\rm hom}(X)_{\bf Q}$ for the group of homologically trivial codimension-$p$ cycles with rational coefficients, and similarly for $q$. Theorem provides a pairing with values in $H^2_{\text{\rm\scriptsize\'et}}(B,{\bf Q}_\ell(1))$. The motivic origin conjecture. This pairing comes from a pairing

$$CH^p_{\rm hom}(X)_{\bf Q}\otimes CH^q_{\rm hom}(X)_{\bf Q}\to {\rm Pic}(B)_{\bf Q}$$

followed by the cycle map. The conjecture is proved when $X$ extends to a smooth proper $B$-scheme; there is also evidence in the abelian-variety case, but the general bad-reduction case remains open.