Given a hyperplane arrangement, one can use the intersection cohomology of a certain toric compactification of the complement to prove the top-heavy conjecture for the lattice of flats (Huh-Wang) and non-negativity of the Kazhdan-Lusztig polynomial (Elias-Proudfoot-Wakefield). In work in progress, we are constructing a combinatorial model for these intersection cohomology groups that should allow us to extend the results to arbitrary (non-realizable) matroids