  A centrally symmetric convex body is a convex compact set with non-empty
interior that is symmetric about the origin. Of particular interest are those
that are both smooth and strictly convex -- known here as regular symmetric
bodies -- since they retain many of the useful properties of the
$d$-dimensional Euclidean ball. We prove that for any given regular symmetric
body $C$, a homothetic packing of copies of $C$ with randomly chosen radii will
have a $(2,2)$-sparse planar contact graph. We further prove that there exists
a comeagre set of centrally symmetric convex bodies $C$ where any
$(2,2)$-sparse planar graph can be realised as the contact graph of a
stress-free homothetic packing of $C$.
