Mokrane's p-adic weight spectral sequence conjecture

Let $E_2^{-r,w+r}$ and $E_2^{r,w-r}$ be the $E_2$-terms of Mokrane's $p$-adic weight spectral sequence for the proper strictly semistable model, and let $K_0$ be the fraction field of $W({\mathbb F}_q)$.

Mokrane's spectral sequence conjecture. For all $r,w$, the map

$$N^r\colon E_2^{-r,w+r}(r)\otimes_{W({\mathbb F}_q)}K_0\xrightarrow{\cong}E_2^{r,w-r}\otimes_{W({\mathbb F}_q)}K_0$$

is an isomorphism.

By the comparison theorem, this is equivalent to the $p$-adic weight-monodromy conjecture. The paper proves the corresponding assertion in its p-adically uniformized setting; it remains open in general.