Chombi,

Just curious but could you describe the cord? Is it like parachute cord? Or, climbing rope in construction? In other words, a woven nylon exterior with a filament nylon core? What is called Kernmantle construction. If so, for what it's worth, back in the days when I was rock climbing, the fuzzing of the outer, woven layer was considered desirable.
It protected the rope from further, deeper abrasion. Not exactly a perfect comparison but, if your cord is nylon, and of this construction, I wouldn't worry about it. Even the cheaper varieties having a tensile strength far exceeding the load your bikes are putting on it.