Kazhdan-Lusztig equivalence for Lie super-algebras

Let $$\mathfrak g$$ be a semi-simple Lie algebra. Kazhdan and Lusztig studied the category of representations of the corresponding affine Lie algebra (the central extension of $$\mathfrak g((t))$$) which are integrable under $$\mathfrak g[[t]]$$; under some assumptions on the central charge they constructed a tensor structure there and proved that the resulting tensor category is equivalent to the category of finite-dimensional representations of the corresponding quantum group (where the parameter $$q$$ in the quantum group depends on the central charge).

My question is the following: is there a known analog of these results for Lie super-algebras? I am particularly interested in $$\mathfrak g=\text{gl}(m|n)$$. Have people studied the Kazhdan-Lusztig category for Lie super-algebras?

In: Kazhdan-Lustzig polynomials and character formulae for the Lie superalgebra $$gl(m|n)$$, J. Amer. Math. Soc. 16 (2002), 185–231, J. Brundan develops a conjecture on the characters for the irreducible modules and tilting modules in the full BGG category $$\mathcal{O}^{m|n}$$ of $$gl(m|n)$$ modules.