Extended Reidemeister move conjecture for knots with discrete diagrams

Define the extended Reidemeister moves to be the standard Reidemeister move set together with a fourth move, in which $A$ is a compact set whose interior remains fixed relative to its boundary; $A$ may contain wild points. Let $f_0,f_1\colon S^1\into\mathbb{R}^3$ be knots admitting discrete diagrams, and suppose that $f_0\cong f_1$. Extended Reidemeister move conjecture. There exists a countable sequence of extended Reidemeister moves satisfying the hypotheses of the ambient-isotopy theorem and taking $f_0$ to $f_1$. This conjecture proposes a countable Reidemeister-type characterization of ambient isotopy for knots admitting discrete diagrams, with the added fourth move allowing compact regions that may contain wild points. The source provides no resolution status.