The homogeneous model-space conjecture for uniform group duals

Let $\Gamma$ fit into a uniform extension

$$\mathbb Z_K^{*M}\to\Gamma\to\mathbb Z_K^M,$$

where uniformity is understood in the strong sense intended in the source. Let $G=\widehat{\Gamma}\subset S_N^+$ be the corresponding group dual, and let $X_G$ be its model space. In the virtually abelian case, let the Haar measure on $X_G$ be the natural invariant measure.

Homogeneous model-space conjecture. Under the stated strong uniformity assumption:

1. The model space $X_G$ is a homogeneous space.
2. In the virtually abelian case, the Haar measure on $X_G$ produces the stationarity of the model.

The conjecture is presented as a refinement of an earlier conjecture. The paper gives no resolution; the phrase “in some strong sense” is deliberately left unspecified in the source.