Irreducibility conjecture for Koopman's representations

Let $\alpha:G\rightarrow\operatorname{Aut}(X)$ be a measurable action of a group $G$ on a measurable space $(X,\mu)$, with $G$-quasi-invariant measure $\mu$. Let

$$\pi^{\alpha,\mu,X}:G\rightarrow U(L^2(X,\mu))$$

be the associated representation, defined by

$$(\pi^{\alpha,\mu,X}_t f)(x)=\left(\frac{d\mu(\alpha_{t^{-1}}(x))}{d\mu(x)}\right)^{1/2}f(\alpha_{t^{-1}}(x)).$$

Write $Z_{\operatorname{Aut}(X)}(\alpha(G))$ for the centralizer of $\alpha(G)$ in $\operatorname{Aut}(X)$, and let $\mu^g$ denote the image of $\mu$ under $g$.

Koopman's irreducibility conjecture. The representation $\pi^{\alpha,\mu,X}$ is irreducible if and only if

1. $\mu^g\perp\mu$ for all $g\in Z_{\operatorname{Aut}(X)}(\alpha(G))\backslash\{e\}$;
2. the measure $\mu$ is $G$-ergodic.

This is presented as a natural generalization of Ismagilov's conjecture. The supplied text gives no resolution, so the conjecture remains open.