Non-degeneracy of swept-area distance for ropelength-filtered knot spaces

Let $K$ be a knot type, let $Y_{\Lambda}(K)$ denote the admissible unit-thickness representatives of ropelength at most $\Lambda$, and let $d_{\Lambda}^{K}$ be the swept-area pseudometric on this space. Consider suitable compact regularity classes of $C^{1,1}$ representatives, and identify representatives under orientation-preserving Euclidean isometries.

Non-degeneracy conjecture. The swept-area pseudometric is non-degenerate on each admissible component of $Y_{\Lambda}(K)$ modulo $\operatorname{Isom}^+(\mathbb{R}^3)$: zero swept-area distance should force equality in the moduli space, not merely equality after passing to the zero-distance quotient.

This is proved in the uniformly non-collinear fixed-$N$ polygonal model, but remains open in the general $C^{1,1}$ ropelength setting.