Asymptotically maximal Schubitopes

We find a layered permutation $w\in S_n$ whose Schubert polynomial ${\mathfrak{S}}_w(x_1,\dots ,x_n)$ has support of size asymptotically at least $n!\mathrm{/}{4}^n$. This gives precise asymptotics for the growth rate of $\beta (n):={\max }_{w\in S_n}\mathrm{\mid }\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}({\mathfrak{S}}_w)\mathrm{\mid }$. We find a different layered permutation $w\in S_n$ whose Grothendieck polynomial has support of size asymptotically at least $n!\mathrm{/}{e}^{\sqrt{2n}\cdot \ln (n)}$ and obtain more precise asymptotics for the growth rate of ${\beta }^{\mathfrak{G}}(n):={\max }_{w\in S_n}\mathrm{\mid }\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}({\mathfrak{G}}_w)\mathrm{\mid }$.