Our 2001 Marquis has a fold sewn into the drape below the pleats and the motor wire runs through that fold. It appears that the wire can be unplugged at the motor end and pulled back through the fold. There is also a plug at the fixed end of the drape so you should unplug it there as well and get the drape off and laying flat and straight before you attempt to pull the wire out. Take it slow as some of the seams may have folds that it will tend to catch on. Work around these by pushing the plug through the fabric using the braille method. It looks like it will be difficult to thread the wire back in, so I would suggest taping a cord to it before you pull it out, pulling the cord in as you pull the wire out, pinning the cord securely on both ends with brass safety pins, clean the drape with the cord in it and use the cord to pull the wire back in after cleaning the drapes.
Just to be clear on this, it is a completely different system from Gerald's. Ours has a motor that moves on the track and stays at the head of the drape pulling the drape behind it. It is sort of like a little model train only it hangs from its track rather than sitting upon it. It chugs from the driver's side to the passenger's side and back, pulling or pushing the drape as appropriate. It also pulls or pushes its power source which is a wire sewn into the drape.
I would not trust this job (the wire) to a cleaner.
Bruce