Quadratic edge-count conjecture for the (1,1) edge-addition process

Let $f_n$ denote the expected number of edges in the task-dependency graph generated by the $(1,1)$ edge-addition process on $n$ vertices.

Quadratic edge-count conjecture.

$$\lim_{n\rightarrow\infty}\frac{f_n}{n^2}\approx 0.37.$$

The source gives that the expectation is $\Theta(n^2)$, with approximate lower and upper bounds of $n^2/4$ and $n^2/2$, and uses experimental results to conjecture a limiting ratio slightly closer to the lower bound. No resolution is provided.