4 problems

The real-rootedness and log-concavity conjecture. One has

The determinant factorization conjecture. Writing $n=2r$ or $n=2r-1$, respectively,

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…

Birkhoff-polytope uniqueness conjecture. If $P(G)$ is combinatorially equivalent to $B_n$ for some $n$, then $G$ is effectively equivalent to $S_n$.