The following question came up in a discussion with a colleague about local Galois representations:

To what extent is the classification of continuous $p$-adic representations of $G_{\mathbf{Q}_{\ell}}$ for $\ell\neq p$ similar to the classification of tamely ramified $p$-adic representations for $\ell=p$?

More precisely, let $\rho: G_{\mathbf{Q}_{\ell}}\rightarrow \mathrm{GL}_n(\mathbf{Q}_p)$ be a (continuous) $p$-adic representation. If $\ell\neq p$, then Grothendieck proved (using the observation that such a representation kills an open subgroup of wild inertia) that $\rho$ is determined by the associated Weil-Deligne representation (see, for example, the notes of Brinon and Conrad, pg. 111, or Taylor's 2002 ICM article).

When $\ell=p$ and $\rho$ is trivial on the wild inertia subgroup, is it the case that $\rho$ is necessarily de Rham?

What seems clear to me is that if one assumes that $\rho$ is Hodge-Tate, then the only Hodge-Tate weight is zero. If indeed $\rho$ were de Rham = pst, then the associated filtered $(\phi,N)$-module would have trivial filtration, and so one ``ought" to be able to recover it from the attached Weil-Deligne representation. In other words, the classification of $p$-adic representations of $G_{\mathbf{Q}_{\ell}}$ for $\ell\neq p$ is literally the same as the case $\ell=p$, provided one throws in the (rather drastic) condition that wild inertia is killed (or at least some open subgroup of it is killed).

Does this sound correct?