Gateva-Ivanova's strong retraction conjecture for squarefree cycle sets

Let $X$ be a finite, nondegenerate cycle set, meaning a set with a binary operation satisfying the cycle-set axioms, and call it squarefree when its diagonal map $T_X:X\to X$, $x\mapsto x\ast x$, is the identity. Call $X$ retractable if there are distinct $x,x'\in X$ such that $x\ast y=x'\ast y$ for every $y\in X$.

Gateva-Ivanova's strong conjecture. If $X$ is a squarefree cycle set with $|X|>1$, then $X$ is retractable.

This conjecture was refuted by Vendramin, who constructed suitable coverings of cycle sets. The broader retraction problem—finding criteria for retractability—remains relevant.