Walkup-type characterization conjecture for the lens space L(3,1)

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

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

$L(3,1)$ $f$-vector conjecture. The $f$-vectors of triangulations of the lens space $L(3,1)$ are characterized by

$$\gamma^*(L(3,1))=\gamma(L(3,1))=18.$$

The paper reports computational verification for vertex numbers from $12$ through $35$, but the complete characterization was left open.