= 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
$$
\boxed{\text{every pairwise disjoint collection of lollipops in the plane is countable}}.
$$
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