Serre's independence and purity conjecture for bad reduction

Let $K$ be a global field, let $X$ be a smooth projective $K$-variety of dimension $n$, and fix $i\in[0,2n]$. For a finite place $v$ and a prime $l$ not dividing the residue characteristic, let $I_v$ be the inertia group, let $\pi_v$ be geometric Frobenius, and let $H^i_l(X)=H^i_{\operatorname{\acute{e}t}}(\overline{X},\mathbf{Q}_l)$. Serre's conjecture. The polynomial

$$\det(1-\pi_v t\mid H^i_l(X)^{I_v})$$

is independent of the choice of $l$ and has integer coefficients; its inverse roots are Weil numbers of weights between $0$ and $i$.

For good reduction, the analogous assertions follow from smooth proper base change and the Weil conjectures. The conjecture concerns the corresponding independence and weight properties at places of bad reduction and is not resolved in the supplied text.