Are equivariant perverse sheaves constructible with respect to the orbit stratification?

Question

[Moved here from MSE]

Consider a variety $X$ over a field $k$ (complex numbers is fine) with the action of a group scheme $G$, and a $G$-equivariant perverse sheaf $F$ over $X$.

Question. Is it true that there exists a stratification $\tau$ of $X$ which is $G$-equivariant and such that $F$ is $\tau$-constructible?

For example, one could inspect the orbit stratification.

I am trying to use the characterization of invariant perverse sheaves as those perverse sheaves such that $act^* F\simeq pr_2^*F$ where $act \colon G\times_k X\to X$ is the action and $pr_2:G\times X\to X$ is the second projection. But I cannot find the solution.

Answer 1

Every complex of sheaves has a unique maximal open subset on which it is locally constant, because if it is locally constant on two open sets, it is locally constant on their union.

Let $U$ be then maximal open subset of $X$ where $F$ is locally constant. Then $U$ is $G$-invariant, because $gU$ is also the maximally open set on which $F$ is locally constant and so $U= gU$.

The complement $X - U$ of $X$ is also $G$-invariant, and $F$ remains $G$-equivariant on restriction to $X - U$. Now we can induct - take the maximal open subset of $X - U$ on which $G$ is locally constant, check it is $G$-invariant, and restrict $F$ to its complement, and so on.