Logarithmic path-length conjecture for the (1,1) edge-removal process

Consider the task-dependency graph generated on $n$ vertices by the $(1,1)$ edge-removal process, and let its maximum directed path length be measured in expectation.

Logarithmic path-length conjecture. The expected maximum directed path length of the resulting task-dependency graph is

$$\Theta(\log n).$$

The conjecture is based on experimental results suggesting logarithmic growth in $n$, in contrast to the apparently linear growth for the edge-addition process. No resolution is provided in the source.