Etale cohomology analogue for the semistable reduction theorem

Let $K$ be a field, $X/K$ a smooth projective variety, $l\neq char(K)$ a prime number and $q\ge 0$. Then we define $\overline{X}:=X_{K_{sep}}$ and denote by $\rho_{X, l}^{(q)}$ the representation of $Gal_K$ on $H^q(\overline{X}, \mathbb{Z}_l)$.

Note that if $X$ is an abelian variety, then $\rho_{X, l}^{(1)}$ is just the representation of $Gal_K$ on the dual of the Tate module $T_l X$ (and $\rho_{X, l}^{(q)}$ is the representation of $Gal_K$ on $\bigwedge^q T_l(X)^\vee$).

Now assume that $S$ is a noetherian integral scheme with function field $K$.

Question 1: Is it true that there is a non-empty open subscheme $U\subset S$ such that $\rho_{X, l}^{(q)}$ factors through $\pi_1(U[1/l])$ for every prime number $l\neq char(K)$?

Remark 1: The answer is yes'', provided $X$ is an abelian variety.

From now on assume that $S=Spec(R)$ with a henselian discrete valuation ring $R$. Then $K$ is the quotient field of $R$. Let $k$ be the residue field of $R$ and $p$ the characteristic of $k$. ($p$ is zero or a prime number.) If $L/K$ is a finite separable extension, then we denote by $I_L\subset Gal_L$ the corresponding inertia group. Let $l\neq p$ be a prime number. Denote by $I_{L, l}$ the maximal pro-$l$ quotient of $I_L$. Then $I_{L, l}$ is procyclic and we choose a generator $g_{L, l}$ of $I_{L, l}$. We say that a representation $\rho: Gal_K\to Aut(T)$ of $Gal_K$ on a finitely generated free $\mathbb{Z}_l$-module $T$ is semistable over $L$, if $\rho$ factors through $I_{L, l}$ and $(\rho(g_{L, l})-Id)^2=0$. Still $X/K$ is a smooth projective variety.

Question 2: Is it true that there is a finite Galois extension $L/K$ such that $\rho_{X, l}^{(q)}$ is semistable over $L$ for every prime number $l\neq p$ and every $q\ge 0$? Is it at least known to be true that there is a finite Galois extension $L/K$ such that $\rho_{X, l}^{(q)}(I_L)$ is a pro-$l$ group for every prime number $l\neq p$ and every $q\ge 0$?

Remark 2: Again the answer is yes'' in the special case where $X$ is an abelian variety. (Cf. SGA 7).

Any answer or hint on the literature would be very helpful; my own search did not yield much.

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For your question $2$, the answer is yes if you weaken your semi-stability condition to require only that $\rho(g_{L,l}-Id)$ is nilpotent. At least when $\mu_{l^\infty}(k)$ is finite, where $k$ is the residue field of $K$, this is a theorem of Grothendieck. See Illusie's article 'Autour de theoreme de monodromie locale', where this is concerned in greater generality. – Keerthi Madapusi Pera Dec 22 '10 at 15:24
Question 1 is affirmative, via standard "spreading out" constructions from EGA IV$_3$ (spread $X$ to smooth proj. over dense open in $S$) and the smooth and proper base change thm. If relax "smooth projective" to "separated and finite type" then it is essentially correct if $S$ is finite type over a field or excellent Dedekind domain, by Deligne's "generic base change" thm (Th. finitude, SGA 4.5), except in principle $U$ may depend on $\ell$. Question 2 is "yes" in general if you define sst "correctly": unipotence of inertia. See Berthelot's Bourbaki expose on deJong's alternations thm. – BCnrd Dec 22 '10 at 21:42
Thanks again for the comment, Brian! Berthelot's article is exactly what I was looking for. – Sebastian Petersen Feb 22 '11 at 17:30