Formulate the wildly ramified local Betti geometric Langlands correspondence

Develop a precise formulation of a wildly ramified local Betti geometric Langlands correspondence involving, on the spectral side, moduli spaces of Stokes data. Specify the objects, stacks, and categories that should participate in this correspondence and articulate the expected relationship between the spectral and automorphic sides in the wild ramification setting.

Background

This paper establishes a monoidal equivalence realizing the tame local Betti geometric Langlands correspondence for reductive groups, thereby settling a conjecture of Ben-Zvi–Nadler in the tame (tamely ramified) setting.

In contrast, the authors note that while it is natural to seek an extension to the wildly ramified setting—where the spectral side should involve moduli of Stokes data—there is currently no precise formulation of such a wildly ramified local Betti correspondence in the literature.