Conditions for parity agreement in finite quandles

Find a condition characterizing when a finite quandle $(X,s)$ satisfies the parity agreement $\#_2(X,s)\equiv\chi^{\mathrm{qdl}}(X,s)\pmod 2$.

Background

The paper constructs finite quandles that fail the parity agreement, even though some examples satisfy it. Thus, parity agreement is not a general property of finite quandles.

The authors leave open the problem of determining conditions under which the two-number and quandle Euler characteristic of a finite quandle have the same parity.

References

Find a condition for a finite quandle $(X,s)$ to satisfy the parity agreement $#_2 (X,s) \equiv \chi{\mathrm{qdl} (X,s)$ ($\mathrm{mod}\, 2$).

— Two-numbers and Euler characteristics for quandles  (2609.05187 - Kai et al., 4 Sep 2026) in Problem 3, final Problem environment