Minimal layered-triangulation conjecture for lens spaces

Let $M$ be a lens space, and call a triangulation minimal if it has the smallest number of tetrahedra among triangulations of $M$. A minimal layered-triangulation of $M$ is a layered-triangulation with the smallest number of tetrahedra among all layered-triangulations of $M$. Minimal layered-triangulation conjecture for lens spaces. With the exceptions of $S^3$, $\mathbb{R}P^3$ and $L(3,1)$, a minimal triangulation of a lens space is a minimal layered-triangulation. The source explains the three exceptions and notes that the asserted relationship is tied to the structure of layered triangulations and the classification of lens spaces; the claim is presented without a resolution in the supplied text.