Suppose you want to construct a representation of an affine algebraic group $G$, you may start with a $G$-equivariant line bundle $\mathcal{L}$ on a $G$-manifold $X$ and then consider global sections, or cohomologies, for example $H^*(X, \mathcal{L})$ becomes a $G$-module.

Suppose now you want to construct a representation of $G$ on a category $\mathcal{C}$ (the type of representations studied in the appendix of [1] or in chapter 7 of [2]). Then you may consider a $G$-equivariant gerbe $\mathcal{G}$ on $X$ and again take global sections.

In the former case you find out that essentially all (finite dimensional, over $\mathbb{C}$, etc) representations of $G$ arise in a geometric way: there exists a manifold $X$ (ie. the flag variety $G/B$ for a choice of a Borel $B \subset G$) and an equivalence of tensor categories between certain $D_X$-modules and $G$-rep. This is known as Beilinson-Bernstein localization and I'll refer to [3] for the precise statements.

My question is if there's a categorical analogue of this statements along the lines of

Is there an equivalence of $2$-categories, between the category of (categorical) representations of $G$ and some category of equivariant gerbes with flat connections on a space $X$ ?

This space might be infinite dimensional. One thing that comes to mind for example is the fact that $D$-modules on the affine grassmanian $Gr_G$ carries a monoidal action of $Rep (G^{\vee})$, and actions of groups on categories are related to these monoidal actions by de-equivariantization [1]. This might be related to my question, but notice that this is not really what I ask up there.

Besides the folklore that "should be true", is there anything concrete written down?

[1] Frenkel and Gaitsgory. Local geometric Langlands correspondence and affine Kac-Moody algebras

[2] Beilinson and Drinfeld Quantization of Hitchin's integrable system and Hecke eigensheaves

[3] Milicic Localization and Representation Theory of Reductive Lie Groups