Radius of comparison and mean dimension

Prove that, for every free minimal dynamical system of an amenable group, the radius of comparison of the crossed product equals half the mean dimension of the action.

Background

The radius of comparison is proposed as an operator-algebraic invariant with a dynamical interpretation through mean dimension. The stated equality is presented as the Phillips–Toms conjecture and is not established in the survey’s generality.

References

Conjecture 6.3.2. (Phillips-Toms). Let G ↷ X be a free, minimal dynamical system of an amenable group G. Then rc(C(X) ⋊ G) = 2 mdim(G ↷ X).

— Classifiability of crossed products  (2610.06586 - Gardella, 5 Oct 2026) in Conjecture 6.3.2, Section 6.3, p. 48