The virtually cyclic intermediary class conjecture

Let $\mathfrak{F}$ be the class of finite or virtually cyclic groups. Kropholler's class ${{\scriptstyle\bf H}}\mathfrak{F}$ and the generalized class ${{\scriptstyle\bf H}}^{\mathfrak{F}}\mathfrak{F}$ are defined by the cellular actions and fixed-point conditions described above. The virtually cyclic intermediary class conjecture.

$${{\scriptstyle\bf H}}\mathfrak{F}={{\scriptstyle\bf H}}^{\mathfrak{F}}\mathfrak{F}.$$

The generalized class is introduced to provide a framework for applying the paper's spectral sequence to Bredon cohomology; the equality would identify it with Kropholler's class for the family of finite or virtually cyclic groups. The source provides no resolution of this proposed equality.