Spectral characterization of stationarity for universal flat models

Let $G\subset S_N^+$ be a transitive closed subgroup. Let $\chi=\sum_i u_{ii}$ be its main character, and let $\mu$ be the distribution of $\chi$. The universal flat model of $C(G)$ is said to be stationary when its associated integration state agrees with the relevant Haar integration state. Spectral stationarity conjecture. The property that the universal flat model of $C(G)$ is stationary is a spectral property of $G$, in the sense that it can be read from $\mu$. This would give a criterion for stationarity in terms of the main-character distribution, extending the known moment and spectral information while leaving the precise criterion to be determined.