Log hard Lefschetz conjecture

Let $H^*$ be one of the log cohomology theories, let $Y$ be a proper strictly semistable log scheme over $k$, and let $\mathcal{L}$ be a line bundle on the underlying scheme $\underline{Y}$. Write $c_1(\mathcal{L})\in H^2(Y)(1)$ for its first Chern class, and suppose that $\underline{Y}$ is proper and geometrically connected of dimension $n$. For $0\leq r\leq n$, cup product defines

$$L^r\colon H^{n-r}(Y)\longrightarrow H^{n+r}(Y)(r).$$

Log hard Lefschetz conjecture. If $\mathcal{L}$ is ample, then $L^r$ is an isomorphism for all $0\leq r\leq n$.

This conjecture extends the hard Lefschetz theorem to proper strictly semistable log schemes and is intended to provide the cohomological duality needed for applications such as Hodge symmetry for rigid varieties. Its resolution status is not specified in the supplied source.