Minimal tetrahedron count for punctured lens spaces

Prove that, for $p>3$, the minimal number of ideal tetrahedra in a triangulation of the punctured lens space $L(p,q)\setminus\{v\}$ equals $\sum_i c_i-3$, where the $c_i$ are the partial denominators of a continued-fraction expansion of $p/q$.

Background

The paper relies on a conjectural formula for the minimal triangulation complexity of a punctured lens space. For $p=2n+3$ and $q=2$, the continued fraction has partial denominators $c_0=n+1$ and $c_1=2$, yielding $n$ tetrahedra. This count is then used to regard the constructed triangulation of $L_{n,1}$ as minimal, and similarly to count $2n$ tetrahedra for $L_{n,n+1}$.