Yep, I tried a few experiments, and Hoofdcommissaris seems to have it right. I turned off the fills and cranked the maximum stroke width all the way up, and it traced it... sort of. There are some strange things where the paths cross, so you'd need to delete the artifacts and do an Average and/or Join on the points to reconnect any broken strokes.

Not sure what your goal is, but if you have a stylus, you might be able to get more brushlike results using Illustrator's paintbrush tool with varying pressure.

Anyway, best luck!