Infinite counterexample conjecture for core edges in lens spaces

A one-vertex triangulation of a lens space is a triangulation with exactly one vertex, and a core edge is an edge that realizes a core curve, whose complement after removing a regular neighbourhood is a solid torus.

Infinite core-edge counterexample conjecture. Every lens space admits infinitely many one-vertex triangulations with no core edges.

The paper proves this for six lens spaces, including the 3-sphere, and reports failed searches for many others. It nevertheless conjectures the statement for every lens space; the supplied text gives no resolution.