Simplicity descent conjecture for differential operator rings

Let $k$ be any field, and let $R\hookrightarrow S$ be an inclusion of $k$-algebras that splits as a map of $R$-modules. Simplicity descent conjecture. If $D_k(S)$ is a simple ring, then $D_k(R)$ is a simple ring. This would explain simplicity for invariant rings through their module-splitting inclusion into a regular ring, independently of more specific features of the group action. Its status in the source is conjectural.