Compact-action infinitesimal containment conjecture

Let $G$ be a compact group with Haar probability measure $\mu$, let $\rho\colon\Gamma\to G$ be a homomorphism, and let $\Gamma$ act on $(G,\mu)$ by $\gamma g=\rho(\gamma)g$. For each conjugacy class of closed subgroups $H\subset G$ with empty interior, let $(Z_H,\tau_H)$ be an auxiliary probability-measure-preserving action. Compact-action infinitesimal containment conjecture.

$$(G,\mu)\xhookrightarrow{\operatorname{inf}}\bigsqcup_{H\subset G}\left[\Gamma/\rho^{-1}(H)\times(Z_H,\tau_H)\right].$$

The conjecture is motivated by obstructions coming from the actions $\Gamma/\rho^{-1}(H)$ and is explicitly described by the authors as more a question than an established conjecture; no resolution is given.