That looks to me like the diamond coating in the clamps got some crystals sticking out on the edges of the teeth and cutting the string when clamping. It often happens on cheaper clamps.
The reason I don’t think it is the gripper as you can see that the damage is happening very close to the frame, exactly were you would clamp the string, the gripper would be much further away from the edge.
What you need to do is figure which clamps in what exact spot do it by inspecting the string and the clamp every time after clamping to pinpoint the spots where it happening. Than you need to take apart the clamp. Make sure NOT to loose the springs and also note how everything is attached. In order to take it apart safely, tighten the jaws of the clamp with your hand when loosening the adjustment screw so that the springs will not jump off when the screw is not holding the jaws together any more. There are at least two springs inside every clamp, some times three. DO NOT LOOSE THEM.
Take a small needle file, or whatever tool you can find and start checking all the teeth around the inside edges for rough spots then cut them away with the file – especially where it corresponds with the damage in the string.
When done, assemble the clamp back same way you took it apart, you will have to tighten the jaws quite a bit to compress the springs before the screw start catching the threads. Good luck!