The generalized weight-monodromy and independence-of-ell conjecture

Let $M$ have weight $w$, let $v\nmid p$ be a finite place of $F$, and let $\sigma(v)$ be a lift of $\operatorname{Fr}(v)$ to $G_{F_v}$. For the monodromy filtration $M_\bullet$ on $M_{\operatorname{et},\mathfrak p}$, consider $\operatorname{Gr}^{M}_{r}M_{\operatorname{et},\mathfrak p}$.

Weight-monodromy and independence-of-ell conjecture. The polynomial

$$\det\left(1-\sigma(v)X\mid\operatorname{Gr}^{M}_{r}M_{\operatorname{et},\mathfrak p}\right)$$

has coefficients in $E$ independently of $\mathfrak p$, and all its roots are Weil numbers of weight $w+r$.

This proposed common generalization asks for simultaneous coefficient independence and the expected purity on every graded piece of the monodromy filtration.