Jannsen's local vanishing conjecture

Let $F$ be a number field, let $\mathfrak{p}$ be a finite place, and let $F_{\mathfrak{p}}$ be the completion of $F$ at $\mathfrak{p}$ with algebraic closure $\overline{F_{\mathfrak{p}}}$. For the relevant étale cohomology representation $H^i_{\operatorname{et}}(\overline{X},\mathbb{Q}_l(n))$, consider the absolute local Galois group $\operatorname{Gal}(\overline{F_{\mathfrak{p}}}/F_{\mathfrak{p}})$. Jannsen's local conjecture. One should have

$$H^2(\operatorname{Gal}(\overline{F_{\mathfrak{p}}}/F_{\mathfrak{p}}),H^i_{\operatorname{et}}(\overline{X},\mathbb{Q}_l(n)))=0$$

if $i+1<n$ or $i+1>2n$. This is the local analogue of the global Jannsen conjecture; the source introduces it as an analogous conjecture and gives no resolution.