For a simple Lie algebra $\mathfrak{g}$, we have the W-algebra of level $k$, denoted by $\mathcal{W}^k(\mathfrak{g})$. Using Wakimoto free field realization and screening operators, Feigin and Frenkel established the celebrated isomorphism $$\mathcal{W}^k(\mathfrak{g})\simeq\mathcal{W}^{^Lk}(^L\mathfrak{g}).$$ Here $^L\mathfrak{g}$ is the Langlands dual of $\mathfrak{g}$, $^Lk\in\mathbb{C}$ satisfies the equation $$r^\vee(k+h^\vee)({^Lk}+{^Lh^\vee})=1,$$ where $r^\vee$ is the lacing number of $\mathfrak{g}$ and $h^\vee$ (resp. $^Lh^\vee$) is the dual Coxeter number of $\mathfrak{g}$ (resp. $^L\mathfrak{g}$).
Qustion: is there a functorial isomorphism here? Namely, can we construct a functor $$F\colon \mathcal{W}^k(\mathfrak{g})\text{-}\mathsf{Mod}\to\mathcal{W}^{^Lk}(^L\mathfrak{g})\text{-}\mathsf{Mod}$$ that is an equivalence? Moreover, since the quantum Drinfeld–Sokolov reduction gives a functor from the category of restricted $\hat{\mathfrak{g}}$-modules of level $k$ to $\mathcal{W}^k(\mathfrak{g})\text{-}\mathsf{Mod}$, is it possible to lift the functor $F$ to a functor between certain $\hat{\mathfrak{g}}$-modules and $\widehat{^L\mathfrak{g}}$-modules?
