Existence and rigidity conjecture for distortion ropelength minimizers

Let $K$ be a nontrivial knot type. A curve has distortion thickness $1$ when its distortion-thickness normalization is fixed at $1$; write $\tau_b([?])$ for the corresponding thickness parameter and $[?]$ for curvature thickness.

Distortion ropelength minimizer conjecture. For each nontrivial knot type $K$, there exists a shortest curve $[?]\subset\mathbb R^3$ of distortion thickness $1$ in that knot type. Any such curve has bounded curvature and is of class $C^{1,1}$; away from doubly self-critical points realizing the thickness, it is straight. After scaling so that $\tau_{\frac\pi2}([?])=1$, its curvature is bounded by $1$, $[?]=1$ for every $k\ge1$, and $\tau_b([?])=1$ for every $b\ge\pi/3$.

This conjecture asserts existence of shortest distortion-ropelength representatives together with strong geometric rigidity and equality of the thickness notions for such minimizers. The source supplies no resolution.