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). The key ingredient is a theory of tensor product categorifications. 

(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. They construct a theory of tensor product categorifications in type A (i.e. existence + uniqueness).  

My question is the following. 

(3) What obstacles arise when extending [Losev-Webster]'s theory to other classical types (i.e. B/C/D)? The paper [Bao-Shan-Wang-Webster](https://arxiv.org/abs/1605.03780) is relevant. 

(4) Can a more general theory, for an arbitrary simple Lie algebra, be developed?  The input data is a Dynkin diagram; the output should be a theory of tensor product categorifications. Williamson's work (using Soergel bimodules) has solved many open problems in the field - perhaps it's possible to use the diagrammatics of Soergel bimodules to solve this problem? 

It'd be a great project for a graduate student (eg. the $\mathfrak{sp}_4$ case).