Phase transition conjecture for the lcm-density of edge ideals

Let $G$ range over all graphs on $n$ vertices, and let the $\lcm$-density refer to the lcm-density of the edge ideal of $G$. Consider the behavior of this quantity across the set of all such graphs as $n$ tends to infinity.

Phase transition conjecture. The $\lcm$-density of edge ideals of graphs exhibits a phase transition phenomenon when considering the set of all graphs on $n$ vertices as $n\to\infty$.

The conjecture is motivated by computations showing low-, transition-, and high-graph-density regimes for graphs on six and seven vertices. The precise nature of the phase transition and its asymptotic behavior are not specified here, so the conjecture remains open.