Existence of length minimizers in fixed distortion classes

Let $[\gamma]$ be a knot type, let $U_C([\gamma])$ be the set of curves in that knot type whose distortion is at most $C$, and let $\delta([\gamma])$ be the distortion thickness of the knot type. Length-minimizer conjecture. For every knot type $[\gamma]$ and every real number $C>\delta([\gamma])$, the set $U_C([\gamma])$ contains a curve minimizing length. This conjecture would resolve the existence issue underlying the paper's main theorem, which assumes a representative of shortest length; the supplied text gives no indication that it has been proved.