$\DeclareMathOperator{\char}{char}\DeclareMathOperator{\gal}{Gal}$ Let $P$ be a smooth projective variety over a field $K$ (one may certainly assume that $K$ is perfect; the case $K=\mathbb{Q}$ already seems to be interesting enough). For some $\ell\neq \char K$, $n>0$, should the $n$-th $\mathbb Q_\ell$-adic Galois cohomology of $X_{K^{sep}}$ be semi-simple as a $\gal(K)$-representation? Certainly; no proof of this fact is known, so I would rather like to know whether it is related with some 'motivic' conjectures.

Some remarks:

For a finite $K$ one can consider the 'motivic' Frobenius; thus the conjecture follows from standard (motivic) ones. Yet this argument does not seem to work already for $K=\mathbb Q$.

It is certainly tempting to apply some polarizability argument. Yet my impression is that polarizability can only be applied to Hodge structures (in general); is this true?

Upd. It seems (see the comment of Ulrich) that 'my conjecture' is wrong for $K= \mathbb Q_\ell$; this settles my question. Yet I wonder where I can find the details for this example (when is the representation corresponding to an elliptic curve with multiplicative reduction indecomposable).

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