 Solution

The conclusion remains true, even though the scaling factor and hence the radius may now be arbitrary. At the point where its handle meets its circle, every lollipop contains a simple triod: take a short arc of the circle in each direction together with a short segment of the handle. Pairwise disjoint lollipops therefore contain pairwise disjoint simple triods. The Moore triod theorem says that every such family in the plane is countable, so