Convergence of annuloids to translating catenoids

Let $s>0$ be fixed, and let $M_i$ be an annuloid in the family $\mathcal A$ from Corollary~, with necksize $s$ and inner width $b_i\to\infty$. After suitable vertical translations, consider the resulting sequence of annuloids. Annuloid–catenoid convergence conjecture. The surfaces $M_i$ converge to the rotationally symmetric translating catenoid $W(s)$, whose neck is a circle of radius $s$. This predicts that annuloids with fixed necksize approach the corresponding rotationally symmetric translating catenoid as their inner width becomes unbounded.