The $cell$-adic monodromy-weight conjecture for $cell\neq p$

Let $K$ be a complete discrete valuation field of characteristic zero with finite residue field $k$ of characteristic $p$, let $W_K$ be its Weil group with valuation map $v$, and let $X$ be a smooth proper scheme purely of dimension $n$ over $K$. For a prime $cell\neq p$, write $H^m_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Q}_{cell})$ for its $cell$-adic étale cohomology, let $N$ be the monodromy operator, and let $\operatorname{Fil}^N_\bullet$ be the associated monodromy filtration. $cell$-adic monodromy-weight conjecture. For $s\in\mathbb{Z}$ and $g\in W_K$, the eigenvalues of $g$ on

$$\operatorname{gr}_s^N H^m_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Q}_{cell})$$

are algebraic numbers and the complex absolute values of their conjugates are $p^{(m+s)v(g)/2}$. The source presents this as a conjecture and does not supply evidence of a general resolution; it is known in several special cases, including good reduction and low cohomological degree.