Freeness conjecture for graded invariant modules over Dedekind domains

Let $R$ be a Dedekind domain, let $G$ be a finite group, and let $G\to \operatorname{GL}_n(R)$ be an $R$-representation. Write $R[x_1,\ldots,x_n]_d^G$ for the degree-$d$ component of the invariant ring. Freeness conjecture. For every $d\in\mathbb{N}$, the $R$-module

$$R[x_1,\ldots,x_n]_d^G$$

is free. This generalizes the preceding conjecture for finite pseudoreflection groups, where $|G|\in R^\times$ and the invariant ring is expected to be a polynomial ring rather than a tensor product of blowup algebras; the source gives no evidence of a resolution.