Willmore minimization conjecture for the bipolar Lawson Klein bottle

Let $\tilde\tau_{3,1}$ be Lawson's bipolar surface, and let $\varphi:\mathbb{S}^4\to\mathbb{R}^4$ be a stereographic projection. For $n\geq 4$, regard $\varphi\circ\tilde\tau_{3,1}$ as an immersion into $\mathbb{R}^n$. Among all immersed Klein bottles in $\mathbb{R}^n$, consider the Willmore energy. Willmore minimization conjecture. The unique minimizer among all immersed Klein bottles in $\mathbb{R}^n$, for $n\geq 4$, is $\varphi\circ\tilde\tau_{3,1}$. This claim is presented as the expected optimality of the bipolar Lawson surface; the surrounding results establish its distinguished spectral and minimal-surface properties, but the global uniqueness and minimization statement remain open.