Global conjecture on the decomposition of Beilinson–Bloch heights

Let $f$ be the family considered above, and let $Z_1$ and $Z_2$ be cycles supported on a single common fiber of $f$. Denote their global Beilinson–Bloch height pairing by $\langle Z_1,Z_2\rangle^{\operatorname{BB}}$, and let $\langle Z_1,Z_2\rangle_v$ be the local pairings defined at each place $v$. Global height decomposition conjecture. The global pairing is the sum of the local pairings:

$$\langle Z_1,Z_2\rangle^{\operatorname{BB}}=\sum_v\langle Z_1,Z_2\rangle_v.$$

This conjecture predicts that the global height of cycles supported on a common fiber decomposes into the local archimedean and non-archimedean contributions defined above. The supplied text gives no evidence of a resolution.