Finiteness conjecture for the V-like Lodha–Moore group

Let $G$ be the Lodha–Moore group and let

$$V(G)=\langle V,G\rangle\leq \operatorname{Homeo}(\{0,1\}^{\mathbb{N}}).$$

Its commutator subgroup is $[V(G),V(G)]$.

Finiteness conjecture. The group $V(G)$ and its commutator subgroup $[V(G),V(G)]$ are of type $\operatorname{F}_\infty$.

The group $V(G)$ is finitely generated, and its commutator subgroup is simple. The methods used in the paper for the analogous groups $\widehat{S}$ and $S$ do not establish this conjecture, so the finiteness properties of $V(G)$ and $[V(G),V(G)]$ remain open.