What's the state of affairs concerning the identification between quantum group reps at root of unity, and positive energy affine Lie algebra reps? 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. 
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[8] D. Kazhdan and G. Lusztig, Tensor structures arising from aﬃne Lie algebras, IV, 
J. Amer. Math. Soc. 7 (1994), 383–453.
 A: The following answer is from Yi-Zhi Huang. I post it here with his permission.

Finkelberg's approach cannot be used to give a
tensor functor in the exceptional case ($\mathfrak g=E_6$, $k=1$,
$\mathfrak g=E_7$, $k=1$ and $\mathfrak g=E_8$, $k=1$ or $2$). Finkelberg's
approach can be sketched as follows:

Kazhdan-Lusztig constructed an equivalence between
a quantum group module category at a root of unity
and an affine Lie algebra module category of a
negative integral level. On the quantum group side, one can take
a subquotient to obtain a semisimple rigid braided tensor category
(in fact a modular tensor category). The same procedure
certainly works on the affine Lie algebra side. So
one also has a semisimple subquotient of the affine Lie
algebra module category of the negative integral level. The main idea
of Finkelberg is to use the contragredient functor for
affine Lie algebra modules. This functor sends modules of
negative levels to those of positive levels (since the functor
is the Lie algebra contragredient functor, not the vertex algebra
contragredient functor). The crucial step is to prove that this
functor is a tensor functor. This is the place where Finkelberg
had a gap. One needs the Verlinde formula to fill the gap.
From this description, you can see that Finkelberg's work
has nothing to do with quantum groups. It is purely a result
on the affine Lie algebra side. It is the work of Kazhdan-Lusztig
that gives the connection with the quantum group.
Now since Kazhdan-Lusztig's work does not cover the exceptional
cases, Finkelberg's approach fails completely. To construct a
tensor functor, one has to do this directly. The abelian categories are
obviously equivalent and the fusion rules are known to be the same.
But these are far from a construction of a tensor category equivalence.
One possibility is to see whether Kazhdan-Lusztig's method can be
adapted. But in this case one has to work directly with the semisimple
subquotient of the quantum group category, since on the affine Lie algebra
side the category is not a subquotient of a nonsemisimple rigid braided tensor
category. It does not seem to be easy to directly adapt the method
of Kazhdan-Lusztig since the representation theory of affine Lie
algebras at positive integral levels are very different from the representation
theory of affine Lie algebras at negative integral levels.
I think that if there is a construction in the exceptional cases (mainly
the case $E_8$, level $2$), it should also work in the general positive level
case. Finkelberg's construction is natural as a construction of an equivalence
between a positive integral level category and a negative integral level category.
But it is not natural as a construction of an equivalence between
a quantum group category and a positive integral level category. The failure
in the exceptional cases is an indication. There should be a direct
and natural construction that works in all the cases and provides a
true understanding of the connections between the modular tensor
categories from quantum groups and from affine Lie algebras.
A: Here is my understanding of the situation. Question 1: yes, the equivalence is known in all cases. Kazhdan-Lusztig work does have some limitations on the level,
so Finkelberg's approach is not applicable. However the categories in question are
easy enough to work with explicitly: for instance $E_8$ at level 1 has just 1 simple object and $E_8$ at level 2 has 3 simple objects and fusion rules of the Ising category. I should note that in these small level cases there is a problem with the loop group side; for example it is non-trivial to show that this category is rigid (Finkelberg deduces this from his equivalence, which is not available for small levels). This could be verified on a case by case basis, but currently we have Huang's proof of Verlinde conjecture which takes care of all cases and much more. 
Question 2: this equivalence is an equivalence of modular tensor categories, so it includes both braided and ribbon structure. The terminology changed since the time of Finkelberg's paper; his "fusion categories" are what is called "ribbon fusion categories" or "premodular categories" nowadays. 
A: I take the freedom to copy here an email of Finkelberg to myself:

Dear Andre,


*

*The braidings and ribbon structures do also match. The braiding is reconstructed from the local systems; the ribbon structure (balance) comes from the action of $L_0$ and can be explicitly computed on any irreducible. Let me know if you need details on this.

*This is a question to Huang and Lepowsky. As far as I remember, they know it for all algebras and levels without exceptions (see the references in my Erratum). My argument needed the rigidity of KL categories which was only established with the above excetpions.
All the best,
Michael.
