Yu and Chen's uniqueness conjecture for the Canham model

Let $g$ denote the genus and let $\boldsymbol{\iota}_0\in(0,1]$ be the reduced area difference parameter for the Canham model. Let

$$\tau=\frac{3}{2^{5/4}\sqrt{\pi}}.$$

A solution is considered up to homothetic transformation. Yu and Chen's conjecture. (i) If $g=0$ and $\iota_0\in(0,1]$, or if $g=1$ and $\iota_0\in(0,\tau]$, then the Canham model has a unique solution given by a surface of revolution. (ii) If $g=1$ and $\iota_0\in[\tau,1)$, then the Canham model has a unique solution defined by the stereographic image in $\mathbb{R}^3$ of the Clifford torus

$$\left\{\frac{1}{\sqrt{2}}[\cos u,\sin u,\cos v,\sin v]^T:u,v\in[0,2\pi]\right\}\subset\mathbb{S}^3.$$

Existence is known in genus zero for all $\iota_0\in(0,1]$ and in genus one for $\iota_0\in[\tau,1)$; the conjecture concerns uniqueness up to homothety, including the asserted surface-of-revolution solution in the remaining genus-one range.