Rump's retraction conjecture for quasilinear cycle sets

Let $A$ be a finite, nondegenerate cycle set whose underlying set is an abelian group, and call it quasilinear when its operation satisfies

$$a \ast (b+c)=a\ast b+(a-b)\ast c.$$

Call $A$ retractable if distinct elements induce the same left translation.

Rump's retraction conjecture. If $A$ is a quasilinear cycle set with $|A|>1$, then $A$ is retractable.

The paper's stated goal is to prove this conjecture, using left braces, the cabling method, and an extended permutation group. Thus the conjecture is solved in the source paper.