Geometric decomposition conjecture for length-minimising thin physical unknots

Let $\mathcal{U}_\tau$ denote the space of physical unknots subject to the thickness parameter $\tau$, with $\tau\in(0,2)$. A length-minimising unknot is a curve minimising length within $\mathcal{U}_\tau$. Its elementary components may include circular arcs of unit radius, straight segments, and Sussmann helices.

Geometric decomposition conjecture. Every length-minimising unknot in $\mathcal{U}_\tau$ for $\tau\in(0,2)$ admits a decomposition into finitely many elementary geometric components: circular arcs of unit radius, straight segments, and Sussmann helices, with the number of elementary components bounded by $O(1/\tau)$.

The conjecture proposes a canonical decomposition that could lead to exact ropelength solutions beyond the round circle, the only exact ropelength currently identified according to the source. Its relationship to curvature-constrained paths is motivated by the work of Dubins and Sussmann, while the general existence and structure of such decompositions remain open.