Before revision:  The main difficulty in proving this result , is overcome by the proof of Lemma 3.5, which states that one is able to cover the s-neighborhood of a Lipschitz curve  in R^2 , with a finite number  balls of radius s, and connect their centers with another Lipschitz curve , \Gamma_\ast, where H^1(\Gamma_\ast) is arbitrarily close to H^1(\Gamma). 
After revision:  The main difficulty in proving this result , is overcome by the proof of Lemma 3.5, which states that one is able to cover the s-neighborhood of a Lipschitz curve in R^2 , with a finite number balls of radius s, and connect their centers with another Lipschitz curve \Gamma_\ast, where H^1(\Gamma_\ast) is arbitrarily close to H^1(\Gamma).