The entanglement-of-formation averaging conjecture for graph density matrices

Let $G$ be a graph with $|V|=pq$ and $m$ edges. For each edge, let $\sigma(k)$ denote the pure density matrix associated with the $k$-th edge. If $\sigma(G)$ is entangled in $\mathbb{C}_{A}^{p}\otimes\mathbb{C}_{B}^{q}$, then Entanglement-of-formation averaging conjecture.

$$E_{F}(\sigma(G))\approx\frac{1}{m}\sum_{k=1}^{m}E_{F}(\sigma(k)).$$

Here $E_F$ denotes entanglement of formation. The source gives this as an observed approximate relation and provides no resolution, so it remains open.