Rump's prime-order classification conjecture for quasilinear cycle sets

Let $A$ be a quasilinear cycle set, with its underlying set an abelian group, and let $p$ be a prime.

Rump's prime-order conjecture. If $A$ has size $p$, then there is an isomorphism of quasilinear cycle sets

$$A\cong \mathbb{Z}_p,$$

where

$$x\ast y=\alpha\cdot y$$

for some fixed $\alpha\in\mathbb{Z}_p\setminus\{0\}$.

The source says this statement follows from the retraction conjecture for quasilinear cycle sets. Since that conjecture is proved in the paper, this prime-order consequence is solved as well.