The relative p-adic monodromy-weight conjecture for a proper SNCL scheme with a relative SNCD

Let $S$ be a $p$-adic scheme, let $X/S_0$ be a proper SNCL scheme with relative SNCD $D$, and let $f\colon X\to S$ be the structural morphism. Let $P$ and $P^D$ be the weight filtrations on $R^qf_{(X,D)/S*}(\mathcal O_{(X,D)/S})$, and let

$$N\colon R^qf_{(X,D)/S*}(\mathcal O_{(X,D)/S})\longrightarrow R^qf_{(X,D)/S*}(\mathcal O_{(X,D)/S})(-1)$$

be the monodromy operator. Relative p-adic monodromy conjecture. If $\overset{\circ}{X}$ is projective over $\overset{\circ}{S}$, then the relative monodromy filtration $M$ with respect to $P^D$ exists and equals $P$. Equivalently, for $q,e,k\in\mathbb N$, the induced morphism

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

is an isomorphism modulo torsion. The conjecture extends the monodromy-weight statement to the bifiltered complex associated with a relative SNCD; the source gives no resolution status.