TITLE: Is there a "universal" cohomology theory for varieties over p-adic fields?
QUESTION [10 upvotes]: Let $K$ be a $p$-adic field, $X$ a smooth proper algebraic variety over $K$, and $0 \le i \le 2 \dim X$. For a prime $\ell \ne p$ one can consider the $\ell$-adic cohomology $H^i(\overline{X}, \mathbb{Q}_\ell)$, and massage this in the usual way (via Grothendieck's abstract monodromy theorem) to get a Weil–Deligne representation of $K$ with coefficients in $\mathbb{Q}_\ell$. For $p$-adic étale cohomology, there is a more complicated construction starting from $H^i(\overline{X}, \mathbb{Q}_p)$ going via Fontaine's $D_{\text{pst}}$ functor. I gather it is conjectured that all of these Weil–Deligne representations are in fact definable over $\mathbb{Q}$, and they should have the same character and thus be isomorphic up to semisimplification; and this is known in some cases.
Is it expected that there should be a "universal" cohomology theory taking values in the category of Weil–Deligne representations over $\mathbb{Q}$, from which all of the above can be obtained by extending scalars? If so, have there been any attempts to construct such a cohomology theory?

REPLY [15 votes]: I'm seeing this old question just now, and simply wanted to remark that the situation may be slightly better.
Namely, enlarging the category of $\mathbb Q$-vector spaces into the larger semisimple $\mathbb Q$-linear tensor category $\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q})$ defined by Kottwitz (see for example the article Representations of the Kottwitz gerbes of Iakovenko for a definition), one expects that there is a (canonical) cohomology theory valued in Weil–Deligne representations with coefficients in $\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q})$. This should recover the other constructions via natural functors to $\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q_\ell})$ for all primes $\ell$. This is a variant of Conjecture 9.5 in my 2018 ICM paper. (Warning: In that paper, I write $\mathrm{Kt}_{\mathbb Q}$ for what I denote $\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q})$ in this answer.)
While the definition of $\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q})$ is as representations of a gerbe $\mathrm{Kt}_{\mathbb Q}$ that is more-or-less explicitly constructed by class field theory — quite similar to the global Weil group —, we still lack a direct linear-algebraic description of $\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q})$, unfortunately.