The layered-permutation conjecture for maximal Newton-polytope support

For $n\geq 1$, let $\theta_w$ denote the number of supports of the Schubert polynomial $\mathfrak{S}_w(x)$, equivalently the number of lattice points in its Newton polytope, and define

$$\beta_n=\max\{\theta_w\colon w\in S_n\}.$$

A layered permutation is a concatenation of decreasing blocks on consecutive intervals. Layered-maximizer conjecture. For $n\geq 1$, the permutations in $S_n$ attaining the maximum value $\beta_n$ are layered permutations. This is equivalent to saying that the permutations whose Newton polytopes have the largest number of lattice points are layered permutations; the source reports verification for $n$ up to $9$.