Real-rootedness of the $Z$-polynomials of sparse paving matroids and strict interlacing for uniform matroids

We prove that the $Z$-polynomial of every sparse paving matroid has only negative real zeros, confirming the real-rootedness conjecture of Proudfoot, Xu, and Young for this class. We also prove that, for each fixed positive corank, the $Z$-polynomials of uniform matroids in consecutive ranks strictly interlace. Our approach is to prove real-rootedness of the corresponding $\gamma$-polynomials for sparse paving matroids and their strict interlacing for uniform matroids, and then transfer both properties to the corresponding $Z$-polynomials.