The weight-monodromy conjecture for proper smooth varieties

Let $K$ be the local field and $F$ its residue field, let $X_K$ be a proper smooth variety over $K$, and let

$$V:= H^w_{\operatorname{ét}}(X_{K^{\operatorname{sep}}},\mathbb{Q}_l).$$

Let $M$ be the monodromy filtration on $V$ and $W$ the weight filtration on $V$. Weight-monodromy conjecture. The monodromy operator $N$ induces an isomorphism

$$N^r \colon E_2^{-r,\,w+r}(r) \xrightarrow{\cong} E_2^{r,\,w-r}$$

for all $r,w$. Equivalently, under the stated hypotheses on the residue field, the filtrations satisfy $M_kV=W_{w+k}V$ for all $k$. This is also called Deligne's conjecture on the purity of the monodromy filtration. Partial results are known, but the conjecture remains open in general, including in mixed characteristic and dimension at least three.