The finite-quandle separation conjecture for knot symmetry classes

Let $\mathcal G$ be the group generated by reversal and mirror operations on knots, and let $\mathcal K$ be a finite set of knots closed under the action of $\mathcal G$. For a finite sequence of finite quandles $\mathcal S=(Q_1,\ldots,Q_n)$, define

$$C(K)=\bigl(\operatorname{Col}_{Q_1}(K),\operatorname{Col}_{Q_2}(K),\ldots,\operatorname{Col}_{Q_n}(K)\bigr).$$

Finite-quandle separation conjecture. There exists such a sequence $\mathcal S$ satisfying, for all $K,K'\in\mathcal K$,

$$C(K')=C(K)\quad\text{if and only if}\quad K'=K\text{ or }K'=mr(K).$$

The weaker implication that equal coloring vectors force $\mathcal G(K')=\mathcal G(K)$ is also stated. The authors verify these claims computationally for various finite knot sets, including knots with at most 12 crossings, but the general finite-set assertion remains open.