Huisken-Yau-type uniqueness for area-constrained Willmore spheres

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.