This is a question about the proofs of Kazhdan-Lusztig's conjectures for category $\mathcal{O}$ using higher representation theory (avoiding Beilinson-Bernstein's geometric localization theory).

Using Bernstein-Frenkel-Khovanov 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 $k$).

Using Losev-Webster and Webster, that statement can be deduced. They construct a theory of tensor product categorifications in type A (i.e. existence + uniqueness).

Can [Losev-Webster] be simplified, so that the connection to Soergel's J.AMS paper (www.ams.org/jams/1990-03-02/S0894-0347-1990-1029692-5/) becomes clear? It would be interesting to construct a theory of tensor product categorifications, with the input data being a Dynkin diagram (i.e. a simple Lie algebra).