Given a finitely generated field $F$ with prime field $k$, we assume $k$ is finite, of characteristic $p$. Fix a prime $\ell$ invertible in $k$.
In the discussion right after [K, Lemma 2.3], the variant
$$H^{*,\star}:=\tilde{\rm H}^*_{\rm cont}(\text{Spec}(F)/\text{Spec}(k),\mathbf{Z}_{\ell}(\star))$$ of Jannsen's continuous étale cohomology of $\text{Spec}(F)$ is defined, for integers $(*,\star)$, with $*$ nonnegative (notation as in loc cit, with $X = \text{Spec}(F)$ and $S = \text{Spec}(k)$).
A more explicit description of $H^{*,\star}$ can be given, as per loc cit, as follows.
Write $F$ as a filtered colimit of the global complete intersection $k$-subalgebras $A$ of $F$ (see [SP, Tag 07BV)], and denote by ${\rm H}_{\rm cont}^*(\text{Spec}(A),\mathbf{Z}_{\ell}(\star))$ the continuous étale cohomology of Jannsen's.
Remark. Here $\mathbf{Z}_{\ell}(\star)$ is the pro-sheaf $\{\mu_{\ell^n}^{\otimes\star}, n\ge 1\}$, as per construction of Jannsen's continuous étale cohomology. See [J] for details on continuous étale cohomology, and [K, $\S$2] for an overview. Roughly, continuous étale cohomology of a scheme $X$ with coefficients in the pro-sheaf $\mathbf{Z}_{\ell}(\star)$ is, degree-wise, an extension of usual $\ell$-adic cohomology of $X$ by a $\lim^1$ module.
Then the $H^{*,\star}$ from above can be written as $$H^{*,\star} = \varinjlim_{k\subset A\subset F}{\rm H}_{\rm cont}^*(\text{Spec}(A),\mathbf{Z}_{\ell}(\star)).$$
Now define $$H^{*,\star}_{\mathbf{Q}} := H^{*,\star}\otimes_{\mathbf{Z}}\mathbf{Q}.$$
Question. Why does $H^{i,p}_{\mathbf{Q}}$ vanish for $i>p+1$, regardless of the transcendence degree of $F$?
Example. When $F$ is a finite extension of $k$, then we get $$H^{*,\star} = {\rm H}^*_{\rm cont}(F,\mathbf{Z}_{\ell}(\star)).$$ Here the right side is continuous cohomology of the field $F$ as in [J, Thm. 3.2]. Since $F$, in this example, is finite, we get that, for all integers $p$, $H^{i,p}$ is zero for all $i\neq 1$, and finite for $i=1$, so $H^{i,p}_{\mathbf{Q}}=0$ for all $i$.
Remark. From [K, Cor.6.10(b)] we can already deduce that, writing $d$ for the transcendence degree of $F$ over $k$, we have $H^{i,p}_{\mathbf{Q}}=0$ for $i\notin [p,p+d+1]$. If $F$ is separable, the same corollary implies the vanishing for $i\notin[p,2p+1]$, since we can arrange every $A$ from above to be a localization of a $k$-smooth subalgebra of $F$, and ${\rm H}^i_{\rm cont}(\text{Spec}(A),\mathbf{Z}_{\ell}(p))$ is finite for $i\notin [p,2p+1]$. The question is asking if the vanishing holds already for $i > p+1$. According to [K, Thm.8.18], this is a consequence of the Tate Conjecture. The Theorem is more precise though, and the question is asking for less: it only asks if this vanishing is already known to hold, and what's an argument/a reference.
References:
[J] U. Jannsen, Continuous étale cohomology. Math. Annalen.
[K] B. Kahn, A sheaf-theoretic reformulation of the Tate conjecture. Preprint.
[SP] Stacks Project.
