Gateva-Ivanova's multipermutation conjecture for finite square-free solutions

Let $X$ be a finite set with $|X|\geq 2$, and let $r:X\times X\to X\times X$ be a nondegenerate involutive square-free set-theoretic solution of the Yang–Baxter equation. Here, nondegenerate means that, writing

$$r(x,y)=(\sigma_x(y),\tau_y(x)),$$

the maps $\sigma_x$ and $\tau_x$ are bijective for every $x\in X$; involutive means that $r^2=\operatorname{id}_{X\times X}$; and square-free means that $r(x,x)=(x,x)$ for every $x\in X$. The retraction $\operatorname{Ret}(X,r)$ is the involutive solution obtained from the natural equivalence relation on $X$, and its iterates are defined by $\operatorname{Ret}^{k}(X,r)=\operatorname{Ret}(\operatorname{Ret}^{k-1}(X,r))$ for $k>1$.

Gateva-Ivanova's conjecture. Every finite nondegenerate involutive square-free set-theoretic solution $r:X\times X\to X\times X$ such that $|X|\geq2$ is a multipermutation solution; that is, $\operatorname{Ret}^{m}(X,r)$ has one element for some positive integer $m$.

This conjecture is presented as one of the main unsolved problems in the study of finite involutive set-theoretic solutions of the Yang–Baxter equation. It predicts that all such square-free solutions arise after finitely many iterations of retraction.