Let $X$ be a smooth proper variety over a number field $K$, $v$ a place of $K$ lying over a prime number $p$, and $V := H^n(X_{\overline{K}};\mathbb{Q}_p)$. Suppose $V$ is unramified at $v$. Then can we prove that $\mathrm{Frob}_v$ has weight $n$ when acting on $V$?

This is immediate from the Weil conjectures if $X$ has good reduction at $v$, and it follows in general by the weight-monodromy conjecture (since then $N=0$). For Abelian varieties, it follows from the Neron-Ogg-Shafarevich Criterion (as then the variety must have good reduction). It would also follow from a Motivic Neron-Ogg-Shafarevich Criterion.

But is it by any chance known in general?

(Also, what about the $\ell=p$ version, where we assume crystalline?)

Let $X$ be a smooth proper variety over a number field $K$, $v$ a place of $K$ lying over a prime number $p \neq \ell$, and $V := H^n(X_{\overline{K}};\mathbb{Q}_{\ell})$. Suppose $V$ is unramified at $v$. Then can we prove that $\mathrm{Frob}_v$ has weight $n$ when acting on $V$?

This is immediate from the Weil conjectures if $X$ has good reduction at $v$, and it follows in general by the weight-monodromy conjecture (since then $N=0$). For Abelian varieties, it follows from the Neron-Ogg-Shafarevich Criterion (as then the variety must have good reduction). It would also follow from a Motivic Neron-Ogg-Shafarevich Criterion.

But is it by any chance known in general?

(Also, what about the $\ell=p$ version, where we assume crystalline?)