Walkup-type characterization conjecture for the 3-torus

For a triangulation of a $3$-manifold with $n$ vertices and $f_1$ edges, define $\gamma(M)$ and $\gamma^*(M)$ by the Walkup inequalities

$$f_1\geq4n+\gamma(M),\qquad \binom{n}{2}\geq f_1\geq4n+\gamma^*(M).$$

3-torus $f$-vector conjecture. The $f$-vectors of triangulations of the $3$-torus $T^3$ are characterized by

$$\gamma^*(T^3)=\gamma(T^3)=45.$$

The claim would determine exactly which edge counts occur for each vertex number, in the sense of Walkup's characterization. The paper verified many cases computationally but left the full characterization open.