The tetrahedron's exact Nieuwland constant conjecture

For a compact convex set $S\subseteq\mathbb{R}^3$, its Nieuwland constant $\mu_S$ is the largest rescaling factor for which a second copy of $S$ can pass straight through a hole in the interior of the first. The tetrahedron's exact Nieuwland constant conjecture. The Nieuwland constant for the tetrahedron is equal to

$$\frac{\sqrt{6}}{1+\sqrt{2}}.$$

The paper gives an explicit passage establishing this value as a lower bound, while exact Nieuwland constants for the non-cube Platonic solids are generally not known; the conjecture asserts that this lower bound is optimal.