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 $d\ge5$, 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 c