The p-adic monodromy-weight conjecture for projective log smooth schemes

Let $\overset{\circ}{S}$ be a $p$-adic formal scheme and let $\overset{\circ}{X}\to\overset{\circ}{S}$ be projective. For a log smooth scheme $Y/S$, let

$$N\colon Ru_{Y/S*}(\mathcal O_{Y/S})\longrightarrow Ru_{Y/S*}(\mathcal O_{Y/S})(-1)$$

be the monodromy operator, and let $P$ denote the weight filtration on $R^qf_{X/S*}(\mathcal O_{X/S})$. $p$-adic monodromy-weight conjecture. For every nonnegative integer $q$ and every $e\in\mathbb N$, the morphism

$$N^e\colon \operatorname{gr}^P_{q+e}R^qf_{X/S*}(\mathcal O_{X/S})\longrightarrow \operatorname{gr}^P_{q-e}R^qf_{X/S*}(\mathcal O_{X/S})(-e)$$

is an isomorphism modulo torsion. This is presented as a $p$-adic analogue and generalization of earlier monodromy-weight conjectures; the source gives no resolution status.