Let $\mathsf{CF}$ be the class of context-free languages. As is well known, $\mathsf{CF}$ is not closed under intersections; the classical counterexample is the intersection of $\{a^mb^mc^n\mid m,n\in\mathbb N\}$ with $\{a^mb^nc^n\mid m,n\in\mathbb N\}$. Can one explicitly describe the closure of $\mathsf{CF}$ under (finite) intersections?
Note that $\mathsf{CF}$ is a subclass of $\mathsf{CS}$, the context-sensitive languages, and that $\mathsf{CS}$ is closed under intersection, so the closure is contained in $\mathsf{CS}$. Then the question boils down to, is every context-sensitive language a finite intersection of context-free languages, or is there a "nice" intermediate class that is closed under intersections?
I find it surprising that the answer to this natural question appears hard to find in the literature.