Mean-dimension bound for the auxiliary crossed-product radius

Establish whether, for every free minimal action of a countable amenable group G on a compact metrizable space X, the auxiliary radius satisfies rc'(C(X),alpha)≤(1/2)mdim(G↷X).

Background

The paper states that either the dynamical-radius inequality or the corresponding crossed-product Phillips–Toms inequality would imply the same upper bound for the auxiliary radius rc'(C(X),alpha).

That auxiliary-radius inequality is separately identified as unresolved, so it is included as a distinct open problem.

References

Either would imply that

rc' (C (X), ) \leq \frac{1}{2} \operatorname{mdim} (),

which is also open.

— The Dynamical Radius of Comparison for C*-Dynamical Systems  (2609.30211 - Asadi-Vasfi et al., 24 Sep 2026) in Final paragraph of Section \ref{Sec_4712_Open}