The positive spun triangulation conjecture for the hyperbolic manifold Vol3

Let $N$ be the closed hyperbolic three-manifold $\textsc{Vol3}=\operatorname{m007}(3,1)$, the third manifold in the SnapPy census of closed oriented hyperbolic three-manifolds. Let $\gamma$ be an embedded closed geodesic in $N$. A pair $(N,\gamma)$ admits a positive spun triangulation if $N-\gamma$ has an ideal triangulation whose tetrahedra have hyperbolic shapes with positive imaginary parts and whose metric completion is isometric to $N$.

The positive spun triangulation conjecture for Vol3. For any embedded closed geodesic $\gamma$, the pair $(N,\gamma)$ does not admit a positive spun triangulation.

The conjecture is supported by the theorem ruling out positive spun triangulations when $\gamma$ is a shortest or second-shortest geodesic, as well as by extensive computer searches. It was first hinted at by Hodgson and Weeks and later asked by Trnkova; the general assertion remains open.