Let (M,g) be a Riemannian 3-manifold that is asymptotic to Schwarzschild. We study the existence of large area-constrained Willmore spheres $$\Sigma\subset M$$ with non-negative Hawking mass and inner radius $$\rho$$ dominated by the area radius $$\lambda$$. If the scalar curvature of $$(M,g)$$ is non-negative, we show that no such surfaces with $$\log \lambda \ll \rho$$ exist. This answers a question of G. Huisken.
