Uniform finiteness conjecture for constrained torsion over number fields

Let $F$ be a field and $n>0$ an integer, and define

$$\mathscr{F}(F,n):=\{K:F\subset K,\ [K:F]=n\}.$$

For a number field $K$, let $\mathscr{A}(K,g,\ell)$ denote the set of $K$-isomorphism classes of $g$-dimensional abelian varieties $A/K$ satisfying the constrained torsion condition $K(A[\ell^\infty])\subseteq\ten$.

Uniform Version. Let $g>0$ and $n>0$. Then there exists a bound $N=N(g,n)>0$ such that $\mathscr{A}(K,g,\ell)=\varnothing$ for any $K\in\mathscr{F}(\mathbb{Q},n)$ and any prime $\ell>N$.

This strengthens the preceding finiteness assertion by making the bound on $\ell$ depend only on the dimension and the degree of the number field, not on the field itself. The supplied text gives no resolution of this uniform version.