Stanley's Ehrhart positivity conjecture for Birkhoff polytopes

The Birkhoff polytope $B_n$ is the convex hull of all $n\times n$ permutation matrices, equivalently the set of real nonnegative $n\times n$ matrices whose row and column sums are all equal to $1$. A polytope is Ehrhart positive when every coefficient of its Ehrhart polynomial is positive. Stanley's conjecture. Birkhoff polytopes are Ehrhart positive. Birkhoff polytopes have been extensively studied through their volumes and Ehrhart polynomials; the conjecture was made by Stanley in a talk, and the paper notes that available data do not establish it in general.