Minimal layered-triangulation conjecture for the solid torus

Let $p/q$ be a reduced rational number with $0\leq p\leq q$, including the exceptional forms $0/1$ and $1/1$. A $p/q$-layered-triangulation is a layered-triangulation of the solid torus extending the one-vertex $p/q$-triangulation on its boundary, and a minimal triangulation means one using the fewest tetrahedra among all triangulations with the specified boundary extension. Minimal layered-triangulation conjecture for the solid torus. The minimal triangulation of the solid torus that extends the $p/q$-triangulation on the boundary is the minimal $p/q$-layered-triangulation of the solid torus. The authors describe this as a compelling conjecture they have been unable to confirm, so the question is whether layered-triangulations realize the minimum possible number of tetrahedra for every prescribed boundary triangulation.