Semisimplicity of étale cohomology representations

Question

Let $K$ be a number field and $G=Gal(\overline{K}/K)$ the absolute Galois group of $K$. Let $\ell$ be a prime number.

Let $A/K$ be an abelian variety. Then the representation of $G$ on $V_\ell(A)$ is semisimple. This is the famous theorem of Faltings (Invent. Math. 73).

Now let $X/K$ be a smooth projective variety and $0\le q\le 2\dim(X)$, and define $\overline{X}=X_{\overline{K}}$.

Question. Is it known that the representation of $G$ on $H^q(\overline{X}, \mathbb{Q}_\ell)$is semisimple?

Remark. The answer is yes for $q=1$, because $H^1(\overline{X}, Q_\ell)$ is dual to $V_\ell(A)$ where $A$ is the Albanese variety of $X$.

I would also be interested in the case where the number field $K$ is replaced by a global function field (say), and $\ell$ is assumed to be coprime to the characteristic.

Answer 1

This semi-simplicity is a part of what is called the Tate conjecture. It is generally believed to be true, but little is known about it outside the case of $H^1$, in either the finite field or global field case. Searching on mathscinet for "Tate conjecture" (or googling) should turn up the relevant literature.

Answer 2

In https://arxiv.org/pdf/1709.04489.pdf, Moonen proves that for finitely generated fields of characteristic $0$, the Tate conjecture (surjectivity of the cycle class map $\mathrm{CH}^r(X) \otimes \mathbf{Q}_\ell \to \mathrm{H}^{2r}(\bar{X},\mathbf{Q}_\ell)^{G_K}$) implies the semisimplicity conjecture.

(For the Tate conjecture, see e.g. http://www.math.columbia.edu/~chaoli/docs/TateConjecture.html or Tate's article in the Motives volume.)