Procure some bungee cord. You will want ~4 mm bungee cord which can be bought online or at some home improvement stores. In my case, I was able to salvage some from a kayak deck rigging kit.

Cut the bungee cord to length. Approximately 16 inches seems to be a good length. You want the bungee to be slightly taught in the natural resting position. A good rule of thumb is that it should be just short of the width between its mounting points.

The old bungee cord used crimped metal to retain the bungee in its clips. I didn't have this readily available, so I melted the ends of the bungee slightly to mushroom the ends out a little bit, then installed a thin zip tie to accomplish the same thing.

You will have to stretch the bungee to clip it in to the other side, and this will temporarily deform the plastic trim until it is reinstalled into the seat back.