The $\overline{\partial}$-Neumann problem is one of the fundamental PDEs in several complex variables. The solution operator, called the $\overline{\partial}$-Neumann operator, $N$ can give information on the inhomogeneous Cauchy-Riemann equation as well as information on the boundary regularity of biholomorphic mappings. One important question is when is $N$ compactBy the work of Catlin and Sibony, it is known that compactness holds if a potential theoretic condition Property ($P$) holds for the boundary of a smooth, pseudoconvex domain. In this talk, we consider a recent construction due to Dall’Ara and Mongodi called the Levi core and show that compactness holds if Property ($P$) holds for the support of the Levi core.