The converse of the closed-manifold theorem for regular CW complexes

Let $X$ be a regular CW complex, and let $X^{(n)}$ denote its $n$-skeleton. Converse of the closed-manifold theorem. If $X=X^{(n)}$, then $X$ is a closed manifold. This would characterize closed manifolds among regular CW complexes by the absence of cells above dimension $n$, complementing the paper's main theorem; the supplied text gives no evidence that the converse is resolved.