Rapoport–Zink weight spectral sequence conjecture

Let $X$ be a proper smooth variety over $K$ with a proper strictly semistable model ${\mathfrak X}$ over ${\mathcal O}_K$. Let

$$E_1^{-r,w+r}\Rightarrow H^w_{\text{\rm \'et}}(X_{\overline K},{\mathbb Q}_l)$$

be the Rapoport–Zink weight spectral sequence, with monodromy operator $N$.

Rapoport–Zink spectral sequence conjecture. For all $r,w$, the induced map on $E_2$-terms is an isomorphism

$$N^r\colon E_2^{-r,w+r}(r)\xrightarrow{\cong}E_2^{r,w-r}.$$

The paper explains that, because the spectral sequence degenerates at $E_2$, this is equivalent to weight-monodromy for $X$. The paper proves the conjecture in its p-adically uniformized setting; it is not established in general.