The bounded-degree torsion sparsity conjecture for abelian schemes

Let $S/K$ be a variety and let $A/S$ be an abelian scheme. Fix an integer $d\geq1$ and a section $\sigma\in A(S)$ of infinite order. Define

$${\mathbf{T}}(d)=\{s\in S(\overline{K})\mid[\kappa(s):K]\leq d\text{ and }\sigma(s)\text{ is torsion in }A_s(\overline{K})\}.$$

Bounded-degree torsion sparsity conjecture. The set ${\mathbf{T}}(d)$ is not Zariski dense in $S$. The paper proves that this conjecture is equivalent to the strong uniform boundedness conjecture and explains that it would follow from the higher-dimensional generalisation of the Silverman–Tate theorem. It remains open in general.