Profinite rigidity of prime knots

Prove that the profinite completion of the fundamental group of a prime knot complement determines the prime knot, in particular for hyperbolic knots, in the sense of the profinite-rigidity conjecture cited in the paper.

Background

The paper situates its link-colouring conjecture within existing questions about profinite rigidity. It notes that profinite rigidity has been established for the figure-eight knot, torus knots, 2-bridge knots, and 3-tangle Montesinos links, while the broader prime-knot and hyperbolic-knot cases remain conjectural in the cited literature.

The statement is included because the authors explicitly say that this rigidity has been conjectured; the paper reports known special cases but does not claim to resolve the general problem.