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 etale cohomology, there is a more complicated construction starting from $H^i(\overline{X}, \mathbb{Q}_p)$ going via Fontaine's $D_{\mathrm{pst}}$ 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?