Effective weight-monodromy conjecture

Question

$\DeclareMathOperator\Gr{Gr}$Let $G$ be the absolute Galois group of a finite extension of $\mathbb{Q}_p$ with inertia subgroup $I$, and let $V$ be an $\ell$-adic representation of $G$. Grothendieck's Monodromy Theorem says that after passing to an open subgroup of $G$, the action of $I$ is unipotent, and moreover may be described by a nilpotent operator $N \colon V \to V$.

There is a monodromy filtration $M_i V$, such that $N M_i V \subseteq M_{i-2}$ and $N^i \colon \Gr^M_i V \to \Gr^M_{-i} V$ is an isomorphism. The Weight-Monodromy Conjecture states that if $V=H^n(X_{\overline{k}};\mathbb{Q}_{\ell})$ for a variety $X$ over a number field $k$, then the Frobenius action on $\Gr^M_i V$ is pure of weight $n+i$ (in particular, the case of good reduction is essentially the Weil conjecture).

Is there a version of this conjecture stating that $\Gr^M_i V = 0$ for $|i| > n$? I can't seem to find that in the literature (e.g., in Conjecture 1.13 of Scholze, or in Deligne's original article).

This is true for abelian varieties, and it seems implied in the literature, but I haven't found a statement like this.

The statement should more generally say that whenever $V$ is an effective motive of weight $n$, the weights are in the interval $[0,2n]$.

Answer 1

You can bound the filtration length (assuming the WM conj) using the weight spectral sequence of Rapoport--Zink. This is a sp seq converging to $H^*(X_{\overline{k}})$; and if the WM conj holds, then the monodromy filtration coincides with the filtration induced by this spectral seq. (There's a nice account of this in Scholl's paper https://www.dpmms.cam.ac.uk/~ajs1005/preprints/weil-preprint1.pdf.) The $E_1$ page of the spectral seq is explicitly given in terms of the components of the special fibre of a semistable model, so you can make explicit what region of the plane they're supported in and you get a bound on the length of the filtration. (I haven't checked whether that bound is the one you want though, the indexing is pretty barbarous.)

Edit. Prompted by Pol's comment, I did the computation: the $E_1^{ij}$ terms are supported in a parallelogram with vertices at $(0, 0), (0, d), (2d, 0), (2d, -d)$. So in particular the filtration on $H^n$ has at most $1 + \min(n, 2d-n)$ nonzero graded pieces for any $n$.