FFRT 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

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

Here FFRT means that the number of isomorphism classes of indecomposable summands of $R^{1/p^r}$, over all $r\geq 1$, is finite. FFRT conjecture for invariant rings. The ring $R$ satisfies FFRT on a Zariski open dense set of fibers of any finitely generated $\mathbb{Z}$-algebra $A$ over which $G$ and $W$ are defined. This conjecture proposes that the finite-$F$-representation-type property, known for invariant rings of linearly reductive groups, extends generically to reductive groups in characteristic zero. The source gives very little evidence and leaves the assertion open.