Extend the single-restriction characterization beyond complexity ≤ 2 for abelian groups with universal transfer

Determine whether the single‑restriction characterization of the maximal compatible transfer system (Conjecture on disklike G‑transfer systems) holds for all disklike transfer systems of abelian groups that contain the universal transfer 1 → G, without any restriction on complexity.

Background

The authors prove the conjectured single‑restriction characterization in the abelian setting for disklike transfer systems of complexity ≤ 2 that contain the universal transfer 1 → G. They report being unable to extend the proof to arbitrary complexity and suggest a potential bridging lemma as a route to a general proof.

Establishing this extension would significantly broaden the scope of the conjecture’s validity in the abelian case and clarify whether the confirmed low‑complexity behavior persists in general.

References

The most general case we were able to prove for the conjecture is: for every abelian group $G$, every disklike transfer system of complexity $\leq 2$ with the universal transfer $1 \to G$ satisfies \cref{conj:maximal compatible disk-like via single restriction}. We were unable to extend this result to transfer systems of arbitrary complexity, but one plausible bridging lemma might be: If $O$ is a (categorical) disklike transfer system for which every transfer system contained in $O$ of complexity $\leq 2$ satisfies \cref{conj:maximal compatible disk-like via single restriction}, then $O$ satisfies \cref{conj:maximal compatible disk-like via single restriction}.

— Maximal compatibility of disklike $G$-transfer systems  (2604.00335 - DeMark et al., 1 Apr 2026) in Remark in Subsection: Disklike Conjectured Characterization