The uniqueness conjecture for good involutions of connected quandles

A quandle is a set equipped with a binary operation satisfying the quandle axioms; it is connected when its inner automorphism group acts transitively, involutory when each right translation is an involution, and good involution when it satisfies the defining compatibility conditions for a good involution. Uniqueness conjecture. Let $R$ be a connected quandle. If $R$ is noninvolutory, then $R$ has at most one good involution. This conjecture is motivated by computational findings and results for particular examples; its general validity remains open.