The retractability and multipermutation conjecture for finite square-free solutions

Let $(X,r)$ be a solution, where $X$ is finite and $(X,r)$ is square-free. A solution is retractable if its retraction is a strictly smaller solution. If $(X,r)$ has finite order $n$, its multipermutation level is denoted by $\mathop{\mathrm{mpl}}(X,r)$.

Retractability and multipermutation conjecture. Every finite square-free solution $(X,r)$ is retractable. Moreover, every finite square-free solution $(X,r)$ of finite order $n$ is a multipermutation solution satisfying

$$\mathop{\mathrm{mpl}}(X,r)<n.$$

This conjecture, attributed in the source to the first author and dating to 2004, predicts that finite square-free solutions can always be reduced by retraction and hence have finite multipermutation complexity.