Ito's torsion analogue of the weight-monodromy conjecture

Let $K$ be a non-archimedean local field with finite residue field $k$ of cardinality $q$ and characteristic $p>0$. Let $X$ be a proper smooth scheme over $K$, let $w$ be an integer, and for all but finitely many primes $\ell\neq p$ let $\{M_{i,\mathbb F_\ell}\}_i$ be the monodromy filtration on $H^w_{\operatorname{\acute et}}(X_{\overline K},\mathbb F_\ell)$. Define

$$\operatorname{Gr}^{M}_{i,\mathbb F_\ell}:=M_{i,\mathbb F_\ell}/M_{i-1,\mathbb F_\ell}.$$

Torsion analogue of the weight-monodromy conjecture. For every integer $i$, there exists a non-zero monic polynomial $P_i(T)\in\mathbb Z[T]$ such that its roots have complex absolute values $q^{(w+i)/2}$ and

$$P_i(\operatorname{Frob})=0$$

on $\operatorname{Gr}^{M}_{i,\mathbb F_\ell}$ for all but finitely many $\ell\neq p$ and every lift $\operatorname{Frob}\in G_K$ of the geometric Frobenius element. The conjecture extends the weight-monodromy prediction to torsion cohomology; the source proves it in several cases, while the general assertion is not resolved.