  A quandle is an algebraic structure whose binary operation is idempotent,
right-invertible and right self-distributive. Right-invertibility ensures right
translations are permutations and right self-distributivity ensures further
they are automorphisms. For finite connected quandles, all right translations
have the same cycle structure, called the profile of the connected quandle.
Hayashi conjectured that the longest length in the profile of a finite
connected quandle is a multiple of the remaining lengths. We prove that this
conjecture is true for profiles with at most five lengths.
