No. You know those things guys like that are an elastic cord with a hook on each end that they can hold stuff together with? There was about 12-18 inches between the safety release and the nylon strap. It stretched. I noticed a lunge rope also had one but it was longer. The horse could probably pull back from the safety strap I would guess another foot.
I know what you are describing - those mini bungy cord things - I don't know what benefit that would be, maybe someone else does. I haven't ever seen a lunge line like that.
Hair, have you seen lines like that? What is the purpose of them?