The idempotent fim measure–stabilizer correspondence for automorphism groups

Let $\mu\in\mathfrak{M}^{\operatorname{inv}}_{\bar m}(\mathfrak{C},M)$ be fim over $M$. Suppose that $\operatorname{Stab}(\mu)=G_{\pi,\mathfrak{C}}$ for a partial type $\pi(\bar x;\bar y)\vdash\bar x\equiv_{\emptyset}\bar y$, where

$$G_{\pi,\mathfrak{C}}=\{\sigma\in\operatorname{Aut}(\mathfrak{C}):\models\pi(\sigma(\bar m);\bar m)\}.$$

The automorphism-group idempotent fim measure conjecture. The following conditions are equivalent: (i) $\mu$ is idempotent with respect to the new convolution product; (ii) $\mu$ 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, idempotent fim measures in $\mathfrak{M}^{\operatorname{inv}}_{\bar m}(\mathfrak{C},M)$ correspond to relatively $\bar m$-type-definable-over-$M$ fim subgroups of $\operatorname{Aut}(\mathfrak{C})$. This generalizes the preceding definable-group problem from groups in the monster model to relatively type-definable subgroups of its automorphism group; the supplied text does not state a resolution.