Monodromy-weight conjecture

Let $\Ubar$ be the geometric generic fibre in a proper flat family over a local base, and let $W_\bullet$ denote the weight filtration on $H^j(\Ubar,\Ql)$. Let

$$N\colon H^j(\Ubar,\Ql)\to H^j(\Ubar,\Ql)(-1)$$

denote the logarithm of monodromy. Monodromy-weight conjecture. For each $r\geq 0$, $N^r$ induces an isomorphism

$$\bar N^r\colon \operatorname{gr}^W_{j+r}H^j(\Ubar,\Ql)\overset{\sim}{\longrightarrow}\operatorname{gr}^W_{j-r}H^j(\Ubar,\Ql)(-r).$$

This conjecture controls the relation between monodromy and weights and is used in the paper to deduce local invariant cycle statements. The source gives no resolution status.