Categories of $D$-modules on a quantized flag manifold

There are two approaches in defining the category of $D$-modules on a quantized flag manifold. One is due to Lunts and Rosenberg based on the $\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}$- construction of the quantized flag manifold, and the other is due to Backelin and Kremnizer using equivariant $D$-modules on the corresponding quantized algebraic group. In this paper we compare the two approaches.