Continuous and discrete knot-type equivalence for quantum knots

Let $K_1$ and $K_2$ be knot $n$-mosaics, and let $\lvert K_1\rangle$ and $\lvert K_2\rangle$ be the corresponding quantum knots. Let $\underset{n}{\approx}$ denote equivalence under the continuous ambient group, and let $\approx$ denote the stabilized continuous knot-type equivalence; let $\underset{n}{\sim}$ and $\sim$ denote the corresponding discrete mosaic equivalences. Quantum knot-type conjecture.

$$\lvert K_1\rangle \underset{n}{\approx} \lvert K_2\rangle \Longleftrightarrow K_1\underset{n}{\sim}K_2\quad\text{and}\quad \lvert K_1\rangle\approx\lvert K_2\rangle \Longleftrightarrow K_1\sim K_2.$$

This asserts that passing from discrete knot mosaics to their quantum states and the connected continuous ambient group introduces no additional equivalences for basis-state mosaics. Its resolution is not supplied in the source material.