Let $k$ be a field, $X$ a smooth $K$-variety and $\ell$ a prime not dividing the characteristic of $K$.
Then one can make sense of continuous $\ell$-adic etale cohomology (in the sense of Jannsen), both of $X$ and of the base-change $X_{\overline{K}}$ to the separable closure, and there's a spectral sequence $$ E^{pq}_2 = H^p(K, H^q_{\mathrm{et}}(X_{\overline{K}}, \mathbf{Q}_\ell)) \Rightarrow H^{p + q}_{\mathrm{et}}(X, \mathbf{Q}_\ell). $$
According to this paper by Nekovar, the spectral sequence is known to degenerate at $E_2$ if $X$ is proper, as a consequence of various deep results of Deligne.
Are there examples of smooth (but non-proper) $X$ where this sequence really doesn't degenerate at $E_2$? I'm particularly interested in the case where the base field $K$ is a number field or a $p$-adic field.