In the appendix of Serre and Tate "Good Reduction of Abelian Varieties" [Annals of Mathematics 88 (1968), 492-517], the authors make the following conjectures.

Suppose that $X$ is a smooth proper variety over a local field (which does not necessarily have good reduction). Write $\rho_\ell$ for the Galois representation given by $H^n_{et}(X, \mathbb{Z}_\ell)$, for some prime $\ell$ not equal to the residue characteristic. Suppose the residue field is finite. Then Serre and Tate conjectured:

(1) The restriction of $\operatorname{Tr} \rho_\ell$ to the inertia subgroup is locally constant, takes values in $\mathbb{Z}$, and is independent of $\ell$.

(2) The characteristic polynomial of $\rho_\ell(\pi)$ has rational coefficients independent of $\ell$, and that its roots have absolute value $q^{-k/2}$ for some $0 \leq k \leq 2n$. (For any choice of $\pi$ of Frobenius element in the Galois group; here, $q$ is the cardinality of the residue field.)

Are these conjectures now proven, or are they still open? And if they are proven, what is a reference for this?