Let $X$ be a stratified variety, and let $i:S\hookrightarrow X$ be the inclusion of a stratum. Let $F := H^k(i^!\operatorname{IC}_X)$. This is a local system on $S$ whose fiber at a point is isomorphic to the degree $k$ compactly supported cohomology of the stalk of $\operatorname{IC}_X$ at that point.
Now let’s assume that $S \cong \mathbb{C}^\times$, and let $V$ be the fiber of $F$ at $1\in S$. Let $\sigma\in\operatorname{Aut}(V)$ be the monodromy map. Then the cohomology of $S$ with coefficients in $F$ is equal to the cohomology of the complex $$H^0(\mathbb{C}^\times) \otimes V\overset{1-f}{\longrightarrow} H^1(\mathbb{C}^\times) \otimes V.$$
Question: Does the map $1-f$ have to be strictly compatible with the weight filtrations on these groups?
We know that $H^0(\mathbb{C}^\times)$ has weight 0 and $H^1(\mathbb{C}^\times)$ has weight 2. This would mean that $1-f$ can only be nonzero if $\operatorname{gr}(V)$ contains two different summands whose weights differ by exactly 2. In particular, if $V$ is pure of some weight, then $1-f$ would have to vanish. This is the conclusion that I want, but I am suspicious; it sounds too strong!