Unduloid conjecture for the three-dimensional cylinder-to-ball transition state

Let $\omega\in((4\pi)^{-1},\pi/6)$ be the fixed volume fraction, and let $c$ and $K$ be the phase-field parameters. An unduloid is a constant-mean-curvature surface of revolution interpolating between a cylinder and a chain of touching balls, with eccentricity $e$. Unduloid conjecture. As $c\to0$ and $K\to\infty$, the transition state from a cylinder to a ball converges in $L^2$ to the indicator function of an unduloid. As $\omega$ increases from $(4\pi)^{-1}$ to $\pi/6$, its eccentricity $e$ increases from $0$ to $1$. The claim is based on numerical transition states and is not resolved in the supplied text.