The idempotent measure conjecture for automorphism-group actions

Let $\bar m$ enumerate a tuple from a small model $M\preceq\mathfrak{C}$, and let $\mu\in\mathfrak{M}^{\operatorname{def}}_{\bar m}(\mathfrak{C},M)$ be fim over $M$. By the cited result, there are a partial type $\pi(\bar x;\bar y)\vdash\bar x\equiv_{\emptyset}\bar y$ and a subgroup $G_{\pi,\mathfrak{C}}$ such that $\operatorname{Stab}_l(\mu)=G_{\pi,\mathfrak{C}}$. Idempotent measure conjecture. The following conditions are conjectured to be equivalent:

1. $\mu$ is idempotent.
2. $\mu$ is the unique left $G_{\pi,\mathfrak{C}}$-invariant measure in $\mathfrak{M}^{\operatorname{inv}}_{\pi(\bar x;\bar m)}(\mathfrak{C},M)$.

In particular, this predicts a correspondence between idempotent fim measures in $\mathfrak{M}^{\operatorname{inv}}_{\bar m}(\mathfrak{C},M)$ and relatively $\bar m$-type-definable over $M$ fim subgroups of $\operatorname{Aut}(\mathfrak{C})$. The conjecture is presented as the automorphism-group analogue and generalization of the definable-group idempotent measure conjecture; the supplied text gives no resolution, so its general status remains open.