$p$-uniformity conjecture for invariant rings

Let $k$ be an algebraically closed field of characteristic zero and let $W$ be a finite dimensional $k$-representation for a reductive group $G$. Put

$$R=\operatorname{Sym}(W)^G.$$

A Frobenius summand means an indecomposable summand of a Frobenius power of a fiber of $R$ in positive characteristic, and $M_i^{\operatorname{Fr}}$ denotes the Frobenius twist of $M_i$. $p$-uniformity conjecture for invariant rings. There exists a finite set $(M_i)_{i=1}^m$ of indecomposable graded $R$-modules such that, for any finitely generated $\mathbb{Z}$-algebra $A$ over which $G$ and $W$ are defined, there is a Zariski open dense subset of $\operatorname{Spec} A$ on which the Frobenius summands of the fibers of $R$ are obtained by reduction of shifts of $(M_i^{\operatorname{Fr}})_{i=1}^m$. This is a characteristic-uniformity statement related to, but weaker in scope than, FFRT; the source says it is the property studied in the paper and leaves the conjecture open.