Homogeneity and stationarity of universal quasi-flat models

Let $G\subset S_N^+$ be quasi-transitive, with all orbits of size $K$, and write $\Gamma=\widehat{G}$. Let $X_G$ be the universal quasi-flat model space and let

$$\pi:C(G)\to M_K(C(X_G))$$

be the corresponding model. The virtual abelianity conjecture. If $\Gamma$ satisfies a suitable virtual abelianity condition, then:

1. $X_G$ is a homogeneous space.
2. The model $\pi$ is stationary.

The conjecture is motivated by Thoma's theorem and explicit verifications, including the classical case. The paper notes that the relevant virtual abelianity condition is not yet defined in the quantum setting; in the classical case it requires an abelian subgroup of finite index. Thus the conjecture remains open.