Maximum hyperplane intersection conjecture for odd-dimensional permutohedra

Let $P_n$ be the permutohedron, and let $H_0$ be its affine hull. For an affine hyperplane $H\ne H_0$, consider the number of vertices in $H\cap P_n$. Maximum intersection conjecture. If $n$ is odd, then every hyperplane different from $H_0$ contains at most $(n-1)!$ points of $P_n$. This bound would explain the lower bound in the preceding covering conjecture for odd $n$; the source presents the assertion as an unproved belief.