Quantum deformation and Langlands duality for relative Lie algebra cohomology

Let $A=\a0\mathbb{C}[z_+,z_-]\otimes\Lambda(\theta_1,\theta_2,\theta_3)$. Write $C^\bullet(\mathfrak g[A],\mathfrak g;\mathbb{C})[[\hbar]]$ for the relative Chevalley--Eilenberg cochain complex, with classical differential $d$. Define the quantum degree by

$$\deg:=2(J_1+J_2+q_1+q_2+q_3),$$

and consider the four conserved combinations $J_1-J_2$, $J_1+q_1$, $J_1+q_2$, and $J_1+q_3$.

Quantum deformation and Langlands duality conjecture. There exists a square-zero deformation

$$d_{\mathrm{quant}}=d+\hbar d_1+\hbar^2d_2+\cdots$$

of the classical relative Chevalley--Eilenberg differential on $C^\bullet(\mathfrak g[A],\mathfrak g;\mathbb{C})[[\hbar]]$ such that $d_{\mathrm{quant}}^2=0$, is homogeneous of degree $+1$ with respect to the quantum degree, and preserves the four conserved combinations. If

$$H_\hbar(\mathfrak g[A],\mathfrak g):=H\!\bigl(C^\bullet(\mathfrak g[A],\mathfrak g;\mathbb{C})[[\hbar]],d_{\mathrm{quant}}\bigr),$$

then, for every Langlands-dual pair $(\mathfrak g,{}^L\!\mathfrak g)$,

$$H_\hbar(\mathfrak g[A],\mathfrak g)\cong H_\hbar({}^L\!\mathfrak g[A],{}^L\!\mathfrak g)$$

as $\mathbb{C}[[\hbar]]$-modules, compatibly with the quantum degree and the four conserved combinations.

This conjecture proposes a quantum correction to the classical relative cohomology in order to repair the mismatch with Langlands duality. The deformation is required to retain the specified charge bookkeeping, while the claimed isomorphism for every Langlands-dual pair remains unproved in the supplied text.