Kato's log $l$-adic monodromy-weight conjecture

Let $\overset{\circ}{X}$ be the underlying scheme of a proper SNCL scheme over $\kappa$, and let

$$N\colon H^q_{\rm ket}(\overline{X}_{\frac{1}{l^{\infty}}},{\mathbb Q}_l)\longrightarrow H^q_{\rm ket}(\overline{X}_{\frac{1}{l^{\infty}}},{\mathbb Q}_l)(-1)$$

be the $l$-adic monodromy operator. Let $P$ denote the filtration induced by the weight spectral sequence.

Kato's log $l$-adic monodromy-weight conjecture. If $\overset{\circ}{X}$ is projective over $\kappa$, then, for every $e\in{\mathbb Z}_{\geq 1}$, $N$ induces an isomorphism

$$N^e\colon {\rm gr}_{q+e}^P H^q_{\rm ket}(\overline{X}_{\frac{1}{l^{\infty}}},{\mathbb Q}_l)\overset{\sim}{\longrightarrow}{\rm gr}_{q-e}^P H^q_{\rm ket}(\overline{X}_{\frac{1}{l^{\infty}}},{\mathbb Q}_l)(-e).$$

This is a log $l$-adic analogue of the monodromy-weight conjecture: it predicts that the monodromy operator satisfies hard Lefschetz-type isomorphisms on the graded pieces of the weight filtration. The source records this as Kato's conjecture; no resolution status is supplied here.