Non-amenable image criterion for infinitesimal containment of compact actions

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 left translation through $\rho$. Non-amenable image criterion. If $\rho(\Lambda)$ is open in $G$ for every non-amenable subgroup $\Lambda$ of $\Gamma$, then

$$(G,\mu)\xhookrightarrow{\operatorname{inf}}\Gamma.$$

This is stated as a special case of the preceding compact-action conjecture and is presented as an open question supported only by the authors' inability to find counterexamples.