Kusner's conjecture on the first Willmore min-max width

Let $\beta_0$ be the min-max width associated with a nontrivial loop in the space of immersions of the round sphere $S^2$ into $\mathbb{R}^3$, equivalently the infimum of the maximal Willmore energy along paths from the standard embedding $\iota$ to its negative. Kusner's conjecture. We have

$$\beta_0=16\pi,$$

and an optimal path is given by a Willmore gradient flow starting from the inversion of Bryant's minimal surface with $4$ embedded ends. This conjecture identifies the first nontrivial Willmore min-max value and predicts the geometric model generating an optimal path; the supplied text gives no resolution status.