The constrained Willmore minimizer conjecture for tori

Let $\mathcal W$ denote the Willmore functional, and fix a conformal class of tori. The $(1,2)$-equivariant constrained Willmore tori are those constrained Willmore tori of orbit type $(1,2)$ and intrinsic period $1$. Constrained Willmore conjecture. The minimizers of $\mathcal W$ among tori in any prescribed conformal class are the $(1,2)$-equivariant constrained Willmore tori of intrinsic period $1$. Numerical experiments support this claim across conformal classes, while the source notes that for sufficiently nonrectangular classes the candidates may have energy at least $8\pi$ and may not be embedded; existence of immersed minimizers is not known in general.