Simplicity of differential operator rings of invariant rings

Let $k$ be a field, let $W$ be a finite-dimensional $k$-representation of a linearly reductive group $G$, and let $R=S(W)^G$ be the ring of invariants in the symmetric algebra $S(W)$. Invariant-ring simplicity conjecture. The ring $D_k(R)$ is simple. This expectation generalizes characteristic-zero results for several classes of invariant rings and is particularly open in positive characteristic.