Tangency graphs of translates of a smooth strictly convex body

Characterize the graphs that arise as tangency graphs of families of pairwise intersecting translates of a given smooth strictly convex body C in R^3.

Background

Diameter graphs in R3 can be viewed as tangency graphs of pairwise intersecting unit balls, and the paper characterizes them as subgraphs of simple non-bipartite quadrangulations of the projective plane. The final open question asks how this characterization changes when unit balls are replaced by translates of an arbitrary smooth strictly convex body.

References

Suppose $C$ is a smooth strictly convex body in $R3$. Which graphs are tangency graphs of families of pairwise intersecting translates of $C$?

— Geometric Realizations with Strong Self-Duality Part II: Diameter Graphs, Reuleaux Polyhedra, and Thrackles  (2610.06340 - Damásdi, 5 Oct 2026) in Section 12, subsection “Tangency graphs of translates,” question