Hayashi's divisibility conjecture for profiles of finite connected racks

Let $X$ be a finite connected rack. Its profile is the common cycle type

$$\lambda=(\lambda_0^{a_0},\lambda_1^{a_1},\dots,\lambda_t^{a_t}),$$

where $\lambda_0<\lambda_1<\cdots<\lambda_t$ are the cycle lengths of the elements of the image of the rack morphism $\Phi\colon X\to\mathsf{Aut}(X)$, with multiplicities $a_0,a_1,\dots,a_t$.

Hayashi's divisibility conjecture. Each $\lambda_s$, for $0\leq s\leq t$, divides $\lambda_t$.

The conjecture concerns the arithmetic structure of profiles of finite connected racks and was originally stated by Hayashi for quandles. The paper proves special cases, while the general divisibility assertion remains unresolved here.