The higher-genus constrained Willmore conjecture for rectangular conformal types

Let $g$ be a genus and let a surface of genus $g$ have a conformal type called rectangular in the sense used for the quotient by the cyclic $(g+1)$-fold symmetry of Lawson's minimal surface $\xi_{g,1}$. Let the constant-mean-curvature analogs of $\xi_{g,1}$ denote the corresponding one-parameter family of embedded constant-mean-curvature surfaces. Higher genus conjecture. The constrained minimizer of the Willmore energy in a rectangular conformal class of genus $g$ is the constant-mean-curvature analog of Lawson's minimal surface $\xi_{g,1}$. The source reports experimental support and notes existence of constrained minimizers in the relevant classes when the Willmore energy is less than $8\pi$, but the general minimizing assertion remains open.