Jannsen's vanishing conjecture for global Galois cohomology

Let $F$ be a number field with algebraic closure $\overline{F}$, let $X$ be a smooth projective variety over $F$, let $p$ be a prime, and let $S$ contain the places above $\infty$ and $p$ and the primes where $X$ has bad reduction. Write $G_S=\operatorname{Gal}(F_S/F)$, where $F_S$ is the maximal extension of $F$ unramified outside $S$, and put $\overline{X}=X\times_F\overline{F}$. Jannsen's conjecture. One should have

$$H^2(G_S,H^i_{\operatorname{et}}(\overline{X},\mathbb{Q}_p(n)))=0$$

if $i+1<n$ or $i+1>2n$. This is a generalization of the weak Leopoldt conjecture and specifies the Tate twists for which vanishing is expected. The paper notes equivalent formulations using $\mathbb{Z}_p$ and $\mathbb{Q}_p/\mathbb{Z}_p$ coefficients, but does not give a resolution of the conjecture.