Variational filtered log p-adic hard Lefschetz conjecture

Let $X/S_0$ be a proper SNCL scheme with structural morphism $f\colon X\longrightarrow S_0\overset{\subset}{\longrightarrow} S$, and suppose that the relative dimension of $\overset{\circ}{X}\longrightarrow\overset{\circ}{S}_0$ is pure of dimension $d$. Let $L$ be a relatively ample line bundle on $\overset{\circ}{X}/\overset{\circ}{S}_0$, and let $\eta=c_{1,\operatorname{crys}}(L)$ be its Chern class in $R^2f_{X/S*}({\cal O}_{X/S})$. Write $P$ for the weight filtration. Variational filtered log $p$-adic hard Lefschetz conjecture. For every $i$, the cup product

$$\eta^i\colon R^{d-i}f_{X/S*}({\cal O}_{X/S})\otimes_{\mathbb Z}{\mathbb Q}\longrightarrow R^{d+i}f_{X/S*}({\cal O}_{X/S})\otimes_{\mathbb Z}{\mathbb Q}$$

is an isomorphism. In fact, it should be an isomorphism of filtered sheaves

$$\eta^i\colon (R^{d-i}f_{X/S*}({\cal O}_{X/S})\otimes_{\mathbb Z}{\mathbb Q},P)\overset{\sim}{\longrightarrow}((R^{d+i}f_{X/S*}({\cal O}_{X/S})\otimes_{\mathbb Z}{\mathbb Q})(i),P),$$

where

$$P_k(R^{d+i}f_{X/S*}({\cal O}_{X/S})\otimes_{\mathbb Z}{\mathbb Q})(i))=P_{k+2i}R^{d+i}f_{X/S*}({\cal O}_{X/S})\otimes_{\mathbb Z}{\mathbb Q}.$$

This is a filtered, relative logarithmic crystalline analogue of the hard Lefschetz theorem. The supplied text attributes the conjecture to the author's earlier work and gives no evidence that it has been resolved.