The converse Latin-quandle conjecture for the quandle polynomial

Let $Q$ be a finite quandle, and let $qp_Q(s,t)$ denote its quandle polynomial. A quandle is Latin when its operation is left-invertible, in addition to the right-invertibility required by the quandle axioms. Latin-quandle conjecture. Every quandle $Q$ with

$$qp_Q(s,t)=|Q|st$$

is Latin. Direct computations show that the converse of the preceding proposition has no counterexamples with cardinality less than $7$; the assertion is otherwise presented as an unproved converse.