A note on cohomological boundedness for $F$-divided sheaves and $\mathcal{D}$-modules

Let $X$ be a smooth proper scheme over an algebraically closed field $k$ in characteristic $p$. In this short note, by interpreting ${\mathcal{D}}_X$-modules as $F$-divided sheaves and establishing a cohomological boundedness property for $F$-divided sheaves, we prove that any ${\mathcal{O}}_X$-coherent ${\mathcal{D}}_X$-module has finite dimensional ${\mathcal{D}}_X$-module cohomology.