Grand uniqueness conjecture for the genus-zero and genus-one Canham problems

Let $v_0$ denote the reduced volume (isoperimetric ratio), with $v_0\in(0,1)$, and consider the Canham problem of minimizing the relevant Willmore-type energy among closed surfaces of genus $g$ subject to $v=v_0$, modulo homothety. The Clifford torus is

$$\left\{\frac{[\cos u,\sin u,\cos v,\sin v]^T}{\sqrt{2}}:u,v\in[0,2\pi]\right\}\subset\mathbb{S}^3,$$

and its surface of revolution representative in $\mathbb{R}^3$ is

$$T_{\sqrt{2}}=\left\{[(\sqrt{2}+\cos v)\cos u,(\sqrt{2}+\cos v)\sin u,\sin v]:u,v\in[0,2\pi]\right\}.$$

Grand uniqueness conjecture. The genus-$g=0$ or $1$ Canham problem with any isoperimetric ratio constraint $v_0\in(0,1)$ has a unique solution up to homothety. Moreover: (i) for $g=0$ and each $v_0\in(0,1]$, the unique solution is a surface of revolution; (ii) for $g=1$ and each $v_0\in\left(0,\frac{3}{2}(2\pi^2)^{-1/4}\right]$, the unique solution is a surface of revolution; and (iii) for $g=1$ and each $v_0\in\left[\frac{3}{2}(2\pi^2)^{-1/4},1\right)$, the unique solution is a stereographic image in $\mathbb{R}^3$ of the Clifford torus in $\mathbb{S}^3$, equivalently a Möbius transformation of $T_{\sqrt{2} }$. At $v_0=1$ and $g=0$, the solution is the round sphere; at $v_0=1$ and $g\geqslant1$, no solution exists by the isoperimetric inequality. Numerical and variational evidence supports uniqueness in the stated genus-zero and genus-one regimes, while a complete proof remains open.