Dynamical radius versus mean dimension

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

Background

The Phillips–Toms philosophy predicts an upper bound for the radius of comparison of crossed products by half the mean dimension. The paper asks for the analogous inequality at the level of the dynamical radius before passing to the crossed product.

The question is posed for free minimal actions of countable amenable groups on compact metrizable spaces and is explicitly left open.

References

Does the inequality \begin{equation}\label{Eq_6923_ModPT} rc (C (X), ) \leq \frac{1}{2} \operatorname{mdim} () \end{equation} always hold?

Eq_6923_ModPT:

rc(C(X),)≤12mdim⁡()rc (C (X), ) \leq \frac{1}{2} \operatorname{mdim} ()

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