The independent packing conjecture for regular symmetric bodies

Let $C$ be a regular symmetric body, meaning a smooth, strictly convex centrally symmetric convex body in the plane that is not a linear transform of a disc. Let $G$ be a $(2,2)$-sparse graph.

Independent packing conjecture. There exists an independent $C$-packing with contact graph $G$.

This would be an analogue for regular symmetric bodies of the conjectured disc-packing result for $(2,3)$-sparse graphs. The supplied text does not state whether the conjecture has been resolved.