Symplectic Balls in Symmetric Polar Products

For every integer n ≥ 2 and every origin-symmetric convex body $K\subset {\mathbb{R}}^n$, we prove that the Gromov width of $\mathrm{int}K\times \mathrm{int}{K}^{\circ }$ is 4. We construct smooth symplectic embeddings of standard balls of every capacity $0<c<4$ into this polar product. Volume preservation then resolves the symmetric Mahler conjecture positively in every dimension.