Finite F-representation type for invariant rings of reductive groups

Let $k$ be an algebraically closed field of characteristic zero, let $G$ be a reductive group, and let $W$ be a finite-dimensional $k$-representation of $G$. The invariant ring $k[W]^G$ is a finitely generated graded $k$-algebra. FFRT conjecture for invariant rings. The ring $k[W]^G$ satisfies finite F-representation type (FFRT): there are only finitely many isomorphism classes of higher Frobenius summands, up to degree shifts, in its positive-characteristic reductions.

The conjecture extends the known result for invariant rings of linearly reductive groups and is motivated by applications to rings of differential operators. Its status in the stated generality is not resolved in the source.