Density conjecture for fixed-isoperimetric-ratio subdivision surfaces

Let $K$ be a closed oriented surface and let $v\in(0,1)$. Write $\operatorname{Imm}_{\mathscr{S}^j}^v$ for the space of immersed Loop subdivision surfaces over $K$ with isoperimetric ratio $v$, and let $\operatorname{Imm}_{W^{2,2}\cap C^1}^v(K)$ be the elements of $\operatorname{Imm}_{W^{2,2}\cap C^1}(K)$ with isoperimetric ratio $v$.

Fixed-ratio density conjecture. For every $v\in(0,1)$,

$$\bigcup_j \operatorname{Imm}_{\mathscr{S}^j}^v$$

is dense in $\operatorname{Imm}_{W^{2,2}\cap C^1}^v(K)$.

Such a density result would provide the approximation needed for numerical treatment of the Canham and Helfrich problems with a fixed isoperimetric-ratio constraint. The paper presents it as unresolved, and notes that proving it is a key numerical-analysis difficulty.