  We obtain in arbitrary codimension a removability result on the order of
singularity of Willmore surfaces realising the width of Willmore min-max
problems on spheres. As a consequence, out of the twelve families of non-planar
minimal surfaces in $\mathbb{R}^3$ of total curvature greater than $-12\pi$,
only three of them may occur as conformal images of bubbles in Willmore min-max
problems.
