2 problems

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Gateva-Ivanova's strong retraction conjecture for squarefree cycle sets
Gateva-Ivanova's strong conjecture. If $X$ is a squarefree cycle set with $|X|>1$ , then $X$ is retractable.
Conjecturerefutedcycle-sets retraction-problem set-theoretic-yang-baxter-equationarXiv posted 2 days ago
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Cancellativity characterization for monoids of finite non-degenerate braided sets
Cancellativity conjecture. The monoid $S(X,r)$ is cancellative if and only if $r$ is involutive.
Conjectureopenbraided-sets monoids set-theoretic-yang-baxter-equationarXiv posted 2 days ago

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