The 2-lobe conjecture for rectangular constrained Willmore tori

For $b\geq 1$, consider tori in $\mathbb{R}^3$ with rectangular conformal type

$$\mathbb{R}^2/(\mathbb{Z}\oplus ib\mathbb{Z}).$$

The relevant candidates are the homogeneous tori $S^1\times S^1(b)$ in the $3$-sphere and the $2$-lobed Delaunay tori in a $3$-sphere, with stereographic projection producing tori in $\mathbb{R}^3$. 2-lobe conjecture. The constrained minimizers of the Willmore energy are the stereographic projections of the homogeneous tori $S^1\times S^1(b)$ for $1\leq b\leq\sqrt{3}$, and the $2$-lobed Delaunay tori for $b>\sqrt{3}$, limiting to a twice-covered equatorial $2$-sphere as $b\to\infty$. The source gives supporting minimization results near the square conformal structure and describes the proposed family as arising from constant-mean-curvature examples; the full conjecture remains unresolved in the supplied text.