In Lawrence-Venkatesh, they proved in Prop 5.3 a beautiful finiteness property for a locus, successfully avoiding the using of Tate conjecture. Note that the introduction of $size_v$ and friendly places aim to deal with non-semi-simple representations (bad point $y$), so if we follow the steps under the Tate conjecture, maybe we can throw these concepts and re-state the finiteness for a larger set. Also, one can see that they do not use the requirement genus $g\geq 2$ in the proof of Prop 5.3.
Since the requirement $d=[\kappa(z)_w:K_v]\geq 8$ guarantees the monodromy group is bigger than $4d^2$, and for $y\equiv y_0(v)$ there are correspondences $\kappa(z)_w\cong\kappa(z_0)_{w_0}$, we can re-state it as
Prop If $X\to Y'\xrightarrow{\pi}Y$ is an Abelian-by-finite family over $K$, which admits a good model over $\mathcal{O}_S$. Also, we assume that it has full monodromy. Fix $v\notin S$. We split $Y(K)$ into finite residue classes $\Omega_v$. For one $\Omega_v$, if there exists $z\in\pi^{-1}(y)$ and $w|v$ for some $y\in\Omega_v$ such that $[\kappa(z)_w:K_v]\geq 8$, then $\Omega_v$ is finite.
It seems that we can easily constuct a family satisfying the latter condition (I do not try it). So for $Y$ with genus 1, we know $Y(K)$ is infinite by Mordell-Weil. Then we can deduce that for such a family, may be the monodromy group is always small. Can we give a proof of this statement directly?
