The small-index conjecture for subgroups of the Lascar strong automorphism group

Let $M$ be a countable saturated structure. For a finite subset $A\subset M$, let $Autf_{A}(M)$ be the subgroup of $Aut_{A}(M)$ generated by

$$\bigcup\left\{Aut_{N}(M)\;;\;A\subseteq N\prec M\right\}.$$

Here $Autf(M)$ denotes the Lascar strong automorphism group of $M$.

Small-index conjecture. If $H$ is a subgroup of $Autf(M)$ with index strictly less than $2^{\aleph_{0}}$, then there exists a finite subset $A\subset M$ such that

$$Autf_{A}(M)\subseteq H.$$

This conjecture characterizes the expected finite-parameter control of subgroups of small index in $Autf(M)$. It is open, even for countable saturated $\omega$-categorical structures, and arose from the observation that all known countable saturated examples failing the small index property have a non-open subgroup of $Gal(T)$ of countable index.