Skein-module Langlands duality for closed oriented 3-manifolds

Let $G$ be a semisimple algebraic group, and let ${^LG}$ denote its Langlands dual group. Let $M$ be a closed, oriented 3-manifold. Suppose that $\Psi \in \mathbb{C}^\times$ is transcendental, and set

$$q=e^{\mathrm{i}\Psi},\qquad {^Lq}=e^{\mathrm{i}{^L\Psi}}.$$

Skein-module Langlands duality conjecture. There is a linear isomorphism

$$\mathrm{Sk}_{G,q}(M) \cong \mathrm{Sk}_{{^LG},{^Lq}}(M).$$

The conjecture is motivated by the specialization of skein modules at $q=1$ to functions on character varieties, and is a three-dimensional, quantum-skein analogue of Langlands duality. The skein modules are known to be finite-dimensional under these assumptions, reducing the assertion at the level of dimensions to an equality of natural numbers; the full canonical isomorphism remains conjectural.