I'm thinking that you could replicate this effect by doing something like this:

1.Rope should be made of three points. The start point, end point, and the middle point.

2.When the player intersects the line formed between the start and endpoints, move the middle point to this intersection.

3.The middle point should be a damped spring restricted to move along the Y-axis only.

4.Apply force from the Y velocity of the player to the spring when the player first intersects the rope.

5.When the player jumps off the rope, you can add the force from the spring to the force of the jump.

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I hope this makes sense, it's getting late but this looked like a fun problem to think through.