Nieuwland number of the tetrahedron

Determine whether the Nieuwland number of the Tetrahedron equals \(\sqrt{6}/(1+\sqrt{2})\).

Background

The paper lists the conjectured value of the Nieuwland number for the Tetrahedron among conjectures from the literature. The paper proves the corresponding conjecture for the Octahedron but does not resolve the Tetrahedron case. In the discussion, the authors state that they are working toward a proof, confirming that this problem remains unresolved in the manuscript.

References

The following conjectures exist in the literature: \begin{align} \nu(\text{Octahedron}) & = 3\sqrt{2}/4 \approx 1.06, \quad , \label{eq:conj:octa} \ \nu(\text{Tetrahedron}) & = \sqrt{6}/(1+\sqrt{2}) \approx 1.015, \quad , \label{eq:conj:tetra} \ \nu(\text{Dodecahedron}) & = \nu(\text{Icosahedron}) = \xi \approx 1.0108, \quad , \label{eq:conj:dod} \end{align}

— The Nieuwland number of the Octahedron  (2609.10788 - Steininger et al., 9 Sep 2026) in Section 1, Introduction