Finiteness-property characterization conjecture for twisted Brin–Thompson groups

Let $G$ be an oligomorphic group of permutations of a set $S$, and let $SV_G$ denote the associated twisted Brin–Thompson group. For $n\in\mathbb{N}$, write $\operatorname{F}_n$ for the corresponding finiteness property. Finiteness-property characterization conjecture. The group $SV_G$ is of type $\operatorname{F}_n$ if and only if all of the following conditions hold:

1. The action of $G$ on $S$ has finitely many orbits of $n$-element subsets.
2. $G$ is of type $\operatorname{F}_n$.
3. For each $1\leq k<n$, the stabilizer in $G$ of any $k$-element subset of $S$ is of type $\operatorname{F}_{n-k}$.

The characterization is stated as known for $n=1$ and is motivated by analogous results for certain wreath products; its validity in general remains open.