The idempotent measure conjecture for definable groups

Let $G=G(\mathfrak{C})$ be a definable group, and let $\mu\in\mathfrak{M}^{\operatorname{def}}_G(\mathfrak{C},M)$ be fim over $M$. Its right stabilizer, denoted $\operatorname{Stab}(\mu)$, is type-definable over $M$. The measure $\mu$ is idempotent measure conjecture. The following conditions are conjectured to be equivalent:

1. $\mu$ is idempotent with respect to definable convolution.
2. $\mu$ is the unique right $G$-invariant, equivalently unique left $G$-invariant, Keisler measure concentrated on $\operatorname{Stab}(\mu)$.

In particular, the conjecture predicts a correspondence between idempotent fim measures in $\mathfrak{M}^{\operatorname{inv}}_G(\mathfrak{C},M)$ and $M$-type-definable fim subgroups of $G(\mathfrak{C})$. It translates the characterization of convolution-idempotent regular measures on locally compact groups as Haar measures on compact subgroups into the model-theoretic setting. It is known for definable groups in stable theories, abelian definable groups, certain NIP groups when the measure is $G^{00}$-invariant, and Dirac measures in rosy theories; the general case remains open.