Hodge-Riemann relations for Potts model partition functions

We prove that the Hessians of nonzero partial derivatives of the (homogenous) multivariate Tutte polynomial of any matroid have exactly one positive eigenvalue on the positive orthant when $0<q\le 1$. Consequences are proofs of the strongest conjecture of Mason and negative dependence properties for $q$-state Potts model partition functions.