Density-minimizing cone singularity

Prove that the cone over \(\bar{\tau}_{1,2}\) in \(\mathbb{R}^4\) is the lowest-density boundary singularity in \(\mathbb{R}^4\).

Background

The paper first records a conjecture that the cone over the Lawson Möbius band is area-minimizing, while noting that a claimed solution to that conjecture appeared during final preparation. It then poses the stated special case concerning the lowest density among boundary singularities, which is presented as a further conjecture.

References

One may then pose a special case of Conjecture \ref{secondArea}: \begin{conj} The cone over $\bar{\tau}_{1,2}$ in $\mathbb{R}4$ is the lowest density boundary singularity in $\mathbb{R}4$. \end{conj}

— On the Willmore energy of Möbius bands  (2609.26745 - Bernstein et al., 22 Sep 2026) in Conjecture, final paragraph of Section 8 (Problems)