Hyperplane covering conjecture for the permutohedron

Let $P_n$ be the permutohedron with vertex set contained in the hyperplane

$$H_0\colon \sum_{i=1}^n x_i={{n+1}\choose 2}.$$

Let $H_1,\ldots,H_m$ be affine hyperplanes different from $H_0$ whose union contains the vertex set of $P_n$. Hyperplane covering conjecture. If $n$ is odd, then $m\ge n$; if $n\ge 4$ is even, then $m\ge n-1$. This conjecture asserts that the explicit covers described in the paper are extremal; its resolution is not provided in the source.