The monodromy–weight conjecture for smooth proper varieties

Let $K$ be the local field, let $X$ be a proper smooth variety over $K$, and let $H^j(X^{\mathrm{ca}},\overline{\mathbb{Q}}_l)$ carry its monodromy filtration $M_\bullet$ and weight filtration $W_\bullet$. Monodromy–weight conjecture. For every $i\in\mathbb{N}$, one has

$$M_i\left(H^j(X^{\mathrm{ca}},\overline{\mathbb{Q}}_l)\right)=W_{i+j}\left(H^j(X^{\mathrm{ca}},\overline{\mathbb{Q}}_l)\right).$$

This predicts that, for smooth proper varieties, the monodromy filtration agrees with the weight filtration after the cohomological shift by $j$. The supplied text gives no resolution status, so the conjecture is recorded as open.