Here's an outline of a proof of Kazhdan-Lusztig conjectures for category O in type A, using higher representation theory (due to Bernstein-Frenkel-Khovanov, Losev-Webster, Stroppel, etc). 

(1) Using [Bernstein-Frenkel-Khovanov] (https://arxiv.org/abs/math/0002087) and generalizations (Sussan, Stroppel-Mazorchuk, etc), the Kazhdan-Lusztig conjectures in type A are equivalent to the following statement: the classes of the simple modules in their categorification correspond to a "dual canonical basis" in a tensor product representation of $\mathfrak{sl}_k$ (for appropriately chosen $k$). 

(2) Using [Losev-Webster] (https://arxiv.org/abs/1303.1336) and [Webster] (https://arxiv.org/abs/1309.3796), that statement can be deduced. 

My question is the following. 

(3) Can this approach be extended to Kazhdan-Lusztig's conjecture in other classical types (i.e. B/C/D)? This paper of [Bao-Shan-Wang-Webster] is relevant: (https://arxiv.org/abs/1605.03780). 

(4) Can this approach be applied to Lusztig's conjecture for dimensions of simple modules for an algebraic group in positive characteristic (via affine Kazhdan-Lusztig polynomials)?