Kusner's classification conjecture for the min-max decomposition

Let ${\vec{\Phi}}_1,\ldots,{\vec{\Phi}}_p,\vec{\Psi}_1,\ldots,\vec{\Psi}_q$ be the Willmore spheres given by the decomposition in equation (12), where the preceding discussion concerns branched Willmore spheres with no first residue. Kusner's classification conjecture. Then

$$p=1,\qquad q=0,$$

and ${\vec{\Phi}}_1$ is the inversion of Bryant's minimal surface with $4$ embedded planar ends. The theorem preceding this statement explains that the earlier conjecture should be interpreted in this more precise way; the supplied text gives no resolution status.