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Let $k$ be a field, $S/k$ a smooth variety with function field $K$ and $U$ a nonempty open subscheme of $S$. For every finite separable extension $E/K$ we denote by $S^E$ (resp. $U^E$) the normalization of $S$ (resp $U$) in $E$. Let $A/U$ be an abelian scheme with generic fibre $A_\eta$. For every prime number $\ell$ different from $char(k)$ and every finite separable extension $E/K$ we consider the representation $\rho_{\ell, E}$ of $\pi_1(U^E)$ on the $\ell$-torsion part $A_\eta[\ell]$. Let $H(E)$ be the kernel of the epimorphism $\pi_1(U^E)\to \pi_1(S^E)$.

Question: Does there exist a finite separable extension $E/K$ such that for every prime number $\ell\neq char(k)$ the group $\rho_{\ell, E}(H(E))$ is generated by its $\ell$-Sylow subgroups?

Remark: It is known that the answer is yes in the special case $dim(S)=1$. This is a consequence of the semistable reduction theorem of Grothendieck.

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