Uniform Rasmussen–Tamagawa conjecture for abelian varieties

Let $F$ be a number field and let $g>0$. For each prime $\ell$, let $\mathscr{A}(F,g,\ell)$ be the set of $F$-isomorphism classes of $g$-dimensional abelian varieties $A/F$ such that $F(A[\ell^\infty])$ is a pro-$\ell$ extension of $F(\mu_{\ell^\infty})$ unramified away from $\ell$. Uniform Rasmussen–Tamagawa conjecture. There exists a constant $C$ depending only on $g$ and the degree of $F/\mathbb{Q}$ such that $\mathscr{A}(F,g,\ell)=\varnothing$ whenever $\ell>C$. This is a uniform strengthening of the finiteness prediction, with the bound independent of the particular number field of the prescribed degree. The supplied text gives no evidence of resolution.