Real-rootedness of Kazhdan--Lusztig polynomials of sparse paving matroids

We prove that every nonconstant Kazhdan–Lusztig polynomial of a sparse paving matroid has only simple negative zeros. For each rank $d\ge 5$, the Kazhdan–Lusztig polynomial of the uniform matroid of rank $d-2$ on $d$ elements strictly interlaces those of all sparse paving matroids of rank $d$ and corank at least two.