Kusner's Klein-bottle conjecture

Prove that the Klein bottle in \(\mathbb{S}^3\) with least possible Willmore energy is the Lawson Klein bottle \(\tau_{1,2}\).

Background

The paper reduces Kusner’s conjecture for Klein bottles containing and symmetric about the great circle to the corresponding Möbius-band characterization problems. The conjecture concerns the global Willmore minimizer among immersed Klein bottles in the round three-sphere.