Strengthened independence of intersection points conjecture

Let $k$ be a field that is not locally finite, let $C$ be a smooth projective curve of genus at least $1$ over $k$, and let $L$ be a very ample line bundle. For a section $s\in H^0(C,L)$, use the closed points $p_i(s)$ and geometric points $\bar p_i(s)$ of $(s=0)$ as above. Let $A\subsetneq\operatorname{Pic}^{\circ}(C)$ be an Abelian subvariety and let $\Gamma\subset\operatorname{Pic}(C)$ be a finitely generated subgroup containing $[L]$.

Strengthened independence conjecture. For “most” sections, the following maps are injections:

$$\left(\bigoplus_{i\in I}\mathbb Z[p_i(s)]\right)\Big/\sum_{i\in I}[p_i(s)]\hookrightarrow \operatorname{Pic}(C)\Big/\langle A(k),\Gamma\rangle$$

and

$$\left(\bigoplus_{i\in\bar I}\mathbb Z[\bar p_i(s)]\right)\Big/\sum_{i\in\bar I}[\bar p_i(s)]\hookrightarrow \operatorname{Pic}(C_{\bar k})\Big/\langle A(\bar k),\Gamma\rangle.$$

The first is the weak form and the second the strong form.

This is explicitly introduced as a stronger variant needed for the threefold case of the paper’s homeomorphism problem. Its status depends on the unresolved meaning of “most” and no proof is supplied.