Uniqueness for the homogeneous Dirichlet Willmore boundary value problem

We give a sufficient condition for curves on a plane or on a sphere such that if these give the boundary of a Willmore surface touching tangentially along the boundary the plane or the sphere respectively, the surface is necessarily a piece of the plane or a piece of the sphere. The condition we require is that the curves bound a strictly star-shaped domain with respect to the Euclidean geometry in the plane and with respect to the spherical geometry in the sphere, respectively.