Let $k$ be a field, $X$ a smooth projective variety over $k$, $\overline{X} := X\times_k {k}^{\rm sep}$ for a separable closure ${k}^{\rm sep}$ of $k$, $\ell$ a prime with $\ell\in k^{\times}$.

> Are the Galois cohomology groups $$H^i(\text{Gal}({k}^{\rm sep}/k),H^j_{\rm ét}(\overline{X},\mathbf{Z}_{\ell}))$$
finite for $i\ge 1$, $j\ge 0$?

I would appreciate to get some references. Thanks