If you have Standard or HD Micro splitters in your sprayer (fan jet), and hold the sprayer tip close to the fringe, you can often "blast" a lot of the soil from the fringe. Start at the kilim and work it towards the fringe tip. You can see the soiling breaking/washing away. You will be saturation cleaning the fringe. Then agitate and rinse/extract in the usual way.
Like others above, I will then treat with an acetic acid solution, short dwell and vacuum-only extract.