Proper compression for closed neighborhoods in bounded-degeneracy graphs

Determine whether there exists a function f(t) such that every hypergraph of closed neighborhoods in a graph of degeneracy t admits a proper sample compression scheme of size f(t).

Background

This paper gives a linear-size (improper) sample compression scheme for closed neighborhoods in graphs of degeneracy t, namely size t + ⌈log(t+1)⌉ + 1, and shows such results are optimal up to constants for improper schemes. Whether similarly bounded proper schemes exist as a function of t is open.

References

Is there a function $f\colon \mathbb{N} \to \mathbb{N}$ such that every hypergraph of closed neighborhoods in a graph of degeneracy $t$ admits a proper sample compression scheme of size $f(t)$?

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