Cadoret–Tamagawa genus-growth conjecture for torsion multisections

Let $k$ be a finitely generated field over $\mathbb{Q}$, let $B$ be a smooth curve over $k$, and let $\pi:\mathcal{A}\to B$ be an abelian scheme with no non-trivial isotrivial abelian subscheme. For a positive integer $n$, let $A[n]^{\times}$ denote the set of torsion points in $A(\overline{K})$ of exact order $n$. For each $x\in A[n]^{\times}$, let $C_x$ be the Zariski closure of $x$ in $\mathcal{A}$, and define

$$g(n)=\min_{x\in A[n]^{\times}}g(C_x).$$

Cadoret–Tamagawa conjecture. One has

$$\lim_{n\to\infty}g(n)=+\infty.$$

This conjecture concerns the increasing complexity of torsion multisections in non-isotrivial abelian schemes and is attributed in the source to Cadoret and Tamagawa. The supplied source does not state whether it is open or resolved.