Let $X$ be a complex variety with contracting $\mathbb{G}_m$ action. Let $i\colon \{x_0\}\to X$ be the inclusion of the fixed point. Then the simplest case of Braden's theorem (which would then seem to go back to Springer) says that if $\mathcal{F}\in D^b_c(X)$ is a constructible complex which is *monodromic* or *weakly* $\mathbb{G}_m$-*equivariant*, then
$$
i^*\mathcal{F}=R\Gamma(\mathcal{F})
$$
so that the stalk at the attracting point is given by global sections.

To obtain the monodromic category, one starts with objects pulled back from the equivariant derived category via $X\to[X/\mathbb{G}_m]$, then adds all objects obtainable by finite iteration of taking cones in $D^b_c(X)$ between such already-obtained objects.

Often people apply the above theorem just to pullbacks of equivariant objects, but beyond that, are there some hypotheses on $X$ that allow easy checking of whether something is monodromic? For example, for which every complex constructible with respect to $X\setminus{x_0},~x_0$ is monodromic?

Of course, I'm also open to answers applying beyond this special case of Braden's theorem.