Note that the curve of a suspended chain or rope does not depend on the amount of gravity; it's the same curve whether gravity is strong or weak. So if the swinging rope were just exposed to a uniform increase in gravity, there would be no difference in shape.

But a swinging jump rope is a bit different - the centripedal force needed to balance the rotation (so called centrifugal force) varies with the radius from the swinging axis. When the rope is at the bottom, each segment is subject to normal gravity plus a value which depends on its distance from the line between the handles. (And at the top or sides of the swinging path, other vector combinations of fixed gravity plus outward "centrifugal" force). This will create a different curve, but it is beyond my math to give you an equation for it.