Positivity of simplicial volume and uniqueness for the twelve-vertex flag three-spheres

We study the twelve-vertex flag triangulation $T_{12}$ of $S^3$ and the Davis $4$-manifold $M(T_{12})$ associated with it. We prove that $T_{12}$ is, up to isomorphism, the unique flag triangulation of $S^3$ with twelve vertices in which every edge is contained in a square; this proves Milizia's Con