Bounded average simplification-length conjecture for 3-manifold triangulations

Let $\sigma_n$ be the average simplification path length for 3-sphere triangulations at level $n$, and let $\pi_n$ be the corresponding average for closed prime orientable 3-manifold triangulations. Bounded average simplification-length conjecture. Both quantities $\sigma_n$ and $\pi_n$ are in $O(1)$; that is, they are bounded above by a constant that does not depend on $n$. This conjecture is supported by the computational results for the available triangulation census, but its asymptotic assertion remains open.