Determine condition (b) for derived defect bases of selected ring spectra

Determine whether the commutative ring spectra MO, K(n), and T(n) satisfy condition (b) in the definition of a G-finite derived defect base for locally finite Artinian groups, given that they satisfy condition (a).

Background

The paper defines a G-finite derived defect base for a commutative ring spectrum R using two conditions. Condition (a) requires the relevant family of finite subgroups to be finite up to G-conjugacy, while condition (b) requires a uniform bound on the number of generation steps needed to construct the cochain algebra C*(BH;R) from cochains associated to the derived defect base, as H ranges over finite subgroups of G.

For locally finite Artinian groups, the authors establish that MO, K(n), and T(n) satisfy condition (a). They explicitly state that it is currently unknown whether these spectra also satisfy the uniform generation condition (b), which would imply that they admit G-finite derived defect bases and allow the weak-descendability results to apply.