Infinite counterexample conjecture for Seifert-fibre edges

A small Seifert fibre space is a Seifert fibre space containing no embedded two-sided incompressible surfaces. A one-vertex triangulation has exactly one vertex, and an edge is isotopic to a Seifert fibre when the embedded curve it realizes is isotopic to a circle fibre. A prism manifold is the class excluded in the statement, alongside lens spaces.

Infinite Seifert-fibre counterexample conjecture. For every small Seifert fibre space that is not a lens space or prism manifold, there exist infinitely many one-vertex triangulations with no edges isotopic to Seifert fibres.

The paper proves the analogous conclusion for two specific small Seifert fibre spaces and reports failed searches for several others. The general statement is proposed as an unanswered question and is not resolved in the supplied text.