Log-concavity of matroid h-vectors and mixed Eulerian numbers

For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain tautological classes in the combinatorial Chow ring $A^\bullet(M)$ arising from Grassmannians. Using mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the h-vector of a matroid complex, improving on an old conjecture of Dawson whose proof was announced recently by Ardila, Denham and Huh.