Kummer-faithfulness conjecture for finite extensions of random fixed fields

Let $K$ be a finitely generated field over $\mathbb{Q}$ and let $e$ be a positive integer. For $almost all$ $\sigma\in G_K^e$, every finite extension of $\overline{K}(\sigma)$ is Kummer-faithful, meaning that for every abelian variety $A$ over the field, the divisible subgroup $A(L)_\mathrm{div}$ is trivial.

Kummer-faithfulness conjecture. Any finite extension of $\overline{K}(\sigma)$ is Kummer-faithful for almost all $\sigma\in G_K^e$.

This conjecture would provide examples of Kummer-faithful fields that need not be sub-$p$-adic. It builds on known results showing that, under the same finite-generation and $e\geq 2$ hypotheses, such finite extensions are AVKF for almost all $\sigma$, while Kummer-faithfulness has been studied in related work; its general assertion remains open.