Uniqueness conjecture for the 12-vertex flag triangulation of the 3-sphere

Let $T_{12}$ be the triangulation referred to in the surrounding discussion. A flag triangulation of $S^3$ has 12 vertices, and every edge is contained in a square.

Uniqueness conjecture. $T_{12}$ is, up to isomorphism, the only flag triangulation of $S^3$ with 12 vertices in which every edge is contained in a square.

The claim concerns the apparent gap in the enumeration of flag triangulations between 8 and 12 vertices and singles out $T_{12}$ among the 12-vertex examples. The source provides no resolution of this conjecture.