Second-smallest systole in bounded-tetrahedron manifolds

Determine whether, for every \(n\geq5\), the manifold \(\mathrm{m125}(0,0)(F_{n+1},F_n)\) realizes the second-smallest systole in \(\Omega_n\), and prove that the quotient of the smallest and second-smallest systoles in \(\Omega_n\) tends to 1 as \(n\to\infty\).

Background

The main systole conjecture identifies one Fibonacci Dehn filling as the presumed minimizer. The authors observe that reversing the gluing pattern produces m125(0,0)(F_{n+1},F_n), which has the second-smallest systole for the tested range n=5,…,11n=5,\dots,11.

The remark explicitly proposes extending this finite pattern to all larger tetrahedron counts and asks for the asymptotic comparison between the first two minima.

References

We conjecture that this generalizes to all $n\geq 5$ and that the quotient of the first and second smallest systole in $\Omega_n$ tends to 1 as $n\to\infty$.

— Verified Length Spectrum and Margulis number for Hyperbolic 3-Manifolds  (2609.26663 - Goerner et al., 22 Sep 2026) in Introduction, Remark immediately following Conjecture 1