Genus-zero singular points of convergence conjecture

Let $N$ be a flat $3$-torus, and let $M_n$ be a sequence of closed constant mean curvature surfaces converging to a limit surface $M_\infty$. Let $q\in N$ be a singular point of convergence, and for sufficiently small $\varepsilon>0$ set

$$\Sigma_n=\overline{B}_N(q,\varepsilon)\cap M_n.$$

Here $C(\Sigma_n)$ denotes the total absolute Gaussian curvature. Genus-zero singular points of convergence conjecture. For $\varepsilon>0$ sufficiently small and $n$ sufficiently large, $\Sigma_n$ is an annulus and

$$C(\Sigma_n)\in (4\pi-\varepsilon,4\pi+\varepsilon).$$

This conjecture describes the catenoidal necks forming near singular points of multiplicity-one convergence; the preceding result gives connectedness and two boundary components, while the asserted annular topology and curvature estimate remain open.