Compactness conjecture for minimizing sequences of subdivision surfaces

Let $K$ be a closed oriented surface, let $g$ be its genus, and let $\mathscr{S}^j$ denote the level-$j$ Loop subdivision space. Let $\operatorname{Imm}_{\mathscr{S}^j}$ be the corresponding space of immersed subdivision surfaces, and let $\mathbf{x}_j$ be any minimizing sequence in the setting of the paper's Willmore minimization theorem. In particular, $\mathbf{x}_j$ may be an approximate $W$-minimizer over $\operatorname{Imm}_{\mathscr{S}^j}$.

Compactness conjecture for minimizing sequences. There is a sequence of Möbius transformations $G_j$ of $\mathbb{R}^3$ such that the surfaces $G_j\circ\mathbf{x}_j(K)$ can be partitioned into subsequences, each converging in Hausdorff distance to some Möbius representative of a genus-$g$ Willmore minimizer. If the genus-$g$ Willmore minimizer is unique up to Möbius transformations, then the whole sequence $G_j\circ\mathbf{x}_j(K)$ converges in Hausdorff distance to some Möbius representative of a genus-$g$ Willmore minimizer.

This is proposed as a strengthening of the paper's subsequential convergence result for discrete minimizers and would make the asymptotic behavior more useful for computation. The second assertion depends on uniqueness of the continuous minimizer, which the paper notes would follow from the generalized Willmore conjecture; the conjecture itself remains open.