Equivalence of extension for torsion relations and invariant relations

Let $G \to \operatorname{GL}(n,\mathbb{C})$ be a representation with discrete kernel and closed image. Let $R$ be a torsion relation for $G$, and let $R'$ be the induced invariant relation for maps to $\operatorname{GL}(n,\mathbb{C})/G$. The torsion-relation extension conjecture. Integral maps of $R'$ extend from every domain $M$ in every Stein manifold to the envelope of holomorphy of $M$ if and only if integral structures of $R$ extend from every such domain to its envelope of holomorphy. This identifies extension of the quotient-map formulation with extension of the corresponding geometric structures; the surrounding proof indicates the two directions are related by passing between a $G$-structure and its quotient map.