I think the guy has the diagram backwards. As the ring moves counter to the boxes motion, it has higher relative velocity (in terms of the box frame), and ther

I think the guy has the diagram backwards. As the ring moves counter to the boxes motion, it has higher relative velocity (in terms of the box frame), and therefore higher mass. As it moves in the direction of motion of the box, it will be moving more slowly (almost sitting still), so it will have less mass / momentum in that direction. The result is that after a few iterations, it will balance out. He's drawn the diagram relative to the outside observer (where the ring seems to go more slowly on the backstroke), but not relative to the box (where the ring has much higher relative velocity on the backstroke, and lower relative velocity on the forward stroke). Another issue here is that the ring uses the box itself to pull itself forward with the spring, and if the mass is increasing, then its pulling ever more on the spring as it "gets faster", thereby slowing the box down by that much on the way back.