A $2$-systolic inequality for ${\mathbb{S}}^2\times P$

We prove a sharp $2$-systolic inequality for four-dimensional products ${\mathbb{S}}^2\times P$, where $P\subset {\mathbb{R}}^2$ is an arbitrary convex polygon. Let $h=g_{{\mathbb{S}}^2}+g_{\mathrm{e}\mathrm{u}}$ be the standard product metric. If a Riemannian metric $g$ on ${\mathbb{S}}^2\times P$ has scalar curvature $\ge \sigma >0$, nonnegative mean curvature on every codimension one face, and dihedral angles no larger than the corresponding dihedral angles of $h$, then both its homotopy and homology $2$-systoles are at most $\frac{8\pi }{\sigma }\mathrm{.}$ This confirms a conjecture of Gromov.