In his paper [1], Finkelberg used Kazhdan-Lusztig's massive work [4,5,6,7,8] to prove that $Rep^{ss}(U_q\mathfrak g)$ (the semisimplification of the category of finite dimensional reps of $U_q\mathfrak g$) is equivalent to the category $Rep_k(\widetilde{L\mathfrak g})$ of level $k$ integrable highest weight modules over the affine Lie algebra $\widetilde{L\mathfrak g}$.

But then, I recently learned (from Section 3 of [3]) that there was an erratum [2] where an error was discovered and corrected, and that there are cases (namely $E_6$, $E_7$, $E_8$ level 1, and $E_8$ level 2) where the Kazhdan-Lusztig story [4,5,6,7,8] cannot be applied...

**Question 1**: Is the equivalence $Rep^{ss}(U_q\mathfrak g)\cong Rep_k(\widetilde{L\mathfrak g})$ known for all $\mathfrak g$ and $k$, or are there exceptions?

**Question 2**: Is the equivalence $Rep^{ss}(U_q\mathfrak g)\cong Rep_k(\widetilde{L\mathfrak g})$ known just at the level of fusion categories, or have the braidings also been compared? How about the ribbon structures?

**References:**

[1] M. Finkelberg, An equivalence of fusion categories, Geom. Funct. Anal. 6 (1996), 249–267.

[2] M. Finkelberg, Erratum to: An equivalence of fusion categories, Geom. Funct. Anal. 6 (1996), 249–267; Geom. Funct. Anal. 23 (2013), 810–811.

[3] Y.-Z. Huang and J. Lepowsky, Tensor categories and the mathematics of rational and logarithmic conformal field theory, ArXiv:1304.7556

[4] D. Kazhdan and G. Lusztig, Aﬃne Lie algebras and quantum groups, Duke Math. J., IMRN 2 (1991), 21–29.

[5] D. Kazhdan and G. Lusztig, Tensor structures arising from aﬃne Lie algebras, I, J. Amer. Math. Soc. 6 (1993), 905–947.

[6] D. Kazhdan and G. Lusztig, Tensor structures arising from aﬃne Lie algebras, II, J. Amer. Math. Soc. 6 (1993), 949–1011.

[7] D. Kazhdan and G. Lusztig, Tensor structures arising from aﬃne Lie algebras, III, J. Amer. Math. Soc. 7 (1994), 335–381. 25

[8] D. Kazhdan and G. Lusztig, Tensor structures arising from aﬃne Lie algebras, IV, J. Amer. Math. Soc. 7 (1994), 383–453.