The Hawking mass of an area-constrained Willmore sphere is a useful quasi-local measure for the strength of the gravitational field of an initial data set for the Einstein field equations. If the scalar curvature satisfies certain asymptotic assumptions, the asymptotic region of such an initial data set is foliated by large area-constrained Willmore spheres with non-negative Hawking mass. It has been conjectured that these are the only such spheres. In this talk, I will present a proof of this conjecture for all large area-constrained Willmore spheres with non-negative Hawking mass and outer radius less than the logarithm of the inner radius. This is joint work with Michael Eichmair, Jan Metzger, and Felix Schulze.