I would replace tied contact with tie constraints. Then you should simplify the model for debugging - e.g. leave only one cord and one or a few selected rollers.
Since contact with the rollers is not working, it’s likely because of too coarse mesh. You have way too elongated elements of the sleeve:
So there are no nodes to engage in contact with the rollers. Or to sufficiently hold the cords.
You should add many more layers to the extruded mesh of the sleeve. Then you can also switch to linear elements.