Ismagilov's irreducibility conjecture for regular representations

Let $G$ be an infinite-dimensional group, let $\widetilde G$ be a larger topological group containing $G$ as a dense subgroup, and let $\mu$ be a measure on $\widetilde G$ that is right or left $G$-quasi-invariant. Write $\mu^{L_t}$ for the image of $\mu$ under left translation and let

$$T^{R,\mu}:G\rightarrow U(L^2(\widetilde G,\mu))$$

be the right regular representation. Here $\mu^{L_t}\perp\mu$ means that the measures are singular, and $G$-ergodicity is understood with respect to the action of $G$.

Ismagilov's conjecture. The right regular representation is irreducible if and only if

1. $\mu^{L_t}\perp\mu$ for all $t\in G\backslash\{e\}$;
2. the measure $\mu$ is $G$-ergodic.

These conditions are necessary for irreducibility, and the conjecture asserts that they are also sufficient. It was verified in many particular cases, but the general case is stated to be an open problem.