Graphs missing a connected partition

We prove that a graph with a cut vertex whose deletion produces at least five connected components must be missing a connected partition of some type. We prove that this also holds if there are four connected components that each have at least two vertices. In particular, the chromatic symmetric function of such a graph cannot be $e$-positive. This brings us very close to the conjecture by Dahlberg, She, and van Willigenburg of non-$e$-positivity for all trees with a vertex of degree at least four. We also prove that spiders with four legs cannot have an $e$-positive chromatic symmetric function.