  In this paper we show a quantitative rigidity result for the minimizer of the
Willmore functional among all projective planes in $\mathbb{R}^n$ with $n\ge
4$. We also construct an explicit counterexample to a corresponding rigidity
result in codimension one, by showing that an Enneper surface might split-off
during a blow-up process. For conformal immersions of spheres with large enough
Willmore energies, we construct explicit counterexamples to a quantitative
rigidity result and this complements the recently obtained rigidity results in
[LaNg13].
