The conjecture that invariant D-modules factor through the abelianization

Let $G$ be a connected linear algebraic group, and let $[G,G]$ denote its commutator subgroup. For a positive integer $n$, write $\sim$ for the relation of $D$-module isomorphism. The quotient map

$$G\twoheadrightarrow G/[G,G]$$

induces a pull-back map on representations.

Invariant D-module abelianization conjecture. The pull-back induces an isomorphism

$$\frac{\{\text{Invariant D-modules of rank } n \text{ over } G\}}{\sim} \cong \frac{\{\text{Invariant D-modules of rank } n \text{ over } G/[G,G]\}}{\sim}.$$

Thus, up to $D$-module isomorphism, invariant D-modules of rank $n$ over a connected linear algebraic group should be determined by those over its abelianization. The conjecture is proposed after examples for reductive and unipotent groups; its general status is not established in the supplied text.