A note on Willmore minimizing Klein bottles in Euclidean space

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We show that Lawson's bipolar surface τ₃,₁ is after stereographic projection the unique minimizer among immersed Klein bottles in its conformal class. We conjecture that it actually is the unique minimizer among immersed Klein bottles into Rⁿ, n≥ 4, whose existence the authors and P. Breuning proved in a previous paper.