If I understand your post correctly, you are not having a problem with the blinds staying up where they are placed, but rather the pleats do not stay folded.
It could be the blinds are too short and they are are flattened too much when extended. Probably unlikely. Maybe the tension is too loose. Maybe try pulling the strings a little tighter. The blinds should offer some resistance when you raise or lower them but not be too difficult to raise.
I don't think restringing them will help.
How old are the blinds? Are they still under warranty? Have they always been like this or have they recently changed?
Edit: I remember reading that someone sprayed starch on the blinds to stiffen the fabric, but I don't remember the details and I have never tried this. Maybe worth trying???
Class C, 2004/5 Four Winds Dutchman Express 28A, Chevy chassis
2010 Subaru Impreza Sedan
Camped in 45 states, 7 Provinces and 1 Territory