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 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 and Webster, 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)? To be precise, the correct set-up is "quantum symmetric pairs", following the paper Bao-Shan-Wang-Webster. Rouquier's theory of tensor product categorifications is less relevant.

(4) Does [Losev-Webster]'s work on tensor product categorification depend in any way on Soergel's diagrammatic theory? If yes, what's the connection?

(5) 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 great to collaborate on this using MathOverflow (i.e. similar to Terry Tao's "polymath"). The $\mathfrak{g}=\mathfrak{sp}_4$ case would be instructive to look at.