The weight-monodromy conjecture for smooth projective varieties

Let $\mathcal{S}/\mathbb{Q}_p$ be a smooth projective scheme, let $E=H^n_{\mathrm{\acute et}}(\mathcal{S}\otimes\overline{\mathbb{Q}_p},\mathbb{Q}_l)$, and let $N:E\to E$ be the nilpotent monodromy operator. Let $M_\bullet$ be the associated increasing monodromy filtration, and let $W^\bullet$ be the decreasing Frobenius weight filtration on $E$, with the weight convention described in the source.

Weight-monodromy conjecture. The two filtrations agree up to the shift specified by

$$\operatorname{Gr}_i^M E=\operatorname{Gr}_{n+i}^W E\qquad\text{for all }i\in\mathbb{Z}.$$

The conjecture compares monodromy and Frobenius weights in the $l$-adic cohomology of a smooth projective variety. The supplied text does not state a resolution in this generality.