Piecewise analyticity conjecture for ropelength minimizers

A ropelength minimizer is a length-minimizing representative of a knot or link type among curves of fixed unit thickness.

Piecewise analyticity conjecture. Ropelength minimizers are piecewise analytic.

Ropelength minimizers are known in the paper to exist and to be $C^{1,1}$, but examples show that they need not be $C^2$. The conjecture concerns the finer structure of their singular sets.