The one-entangled-edge and common-vertex conjecture for graph density matrices

Let $G$ be a graph with $|V|=pq$, and let $entangled edges$ mean edges whose associated pure states are entangled. One-entangled-edge and common-vertex conjecture. If $G$ has only one entangled edge, then $\sigma(G)$ is entangled; if all the entangled edges of $G$ are incident to the same vertex, then $\sigma(G)$ is entangled.

This proposes graph-theoretic sufficient conditions for entanglement of the density matrix associated with a graph. The source provides no resolution, so the conjecture remains open.