Constant-size proper compression for balls in planar graphs

Determine whether there exists a universal constant c such that, for every planar graph G, the hypergraph of balls in G admits a proper sample compression scheme of size c.

Background

Planar graphs have distance VC-dimension at most 4, implying bounded-size (improper) compression via Moran–Yehudayoff. Prior work built proper schemes for closed neighborhoods and, in this paper, for balls of bounded radius O(r log r), but a constant-size proper scheme for all radii remains unknown.

References

Is there a constant $c$ such that for every planar graph $G$, the hypergraph of balls in $G$ admits a proper sample compression scheme of size $c$?

— Sample compression schemes for balls in structurally sparse graphs  (2604.02949 - Bourneuf et al., 3 Apr 2026) in Section 7 (Open problems)