The seminal works of Kapustin—Witten and Gaiotto—Witten propose that various equivalences falling under the umbrella term "Langlands correspondence" can be interpreted as S-duality for a family of topological twists in 4D gauge theory. In this talk, we discuss a program involving joint work in progress with Emilio Franco, Enya Hsiao, and Mengxue Yang, in which we explore and realize some features of this hypothetical Langlands TQFT in the Dolbeault twist. Our constructions are directly motivated by the relative Langlands program of Ben-Zvi—Sakellaridis--Venkatesh and we recover conceptual reformulations of a key construction in higher Teichmüller theory.