Cancellativity characterization for monoids of finite non-degenerate braided sets

Let $(X,r)$ be a finite non-degenerate braided set with 2-cancellation, meaning that the associated cancellation condition holds for pairs of factors. Let $S(X,r)$ be its associated monoid.

Cancellativity conjecture. The monoid $S(X,r)$ is cancellative if and only if $r$ is involutive.

The statement concerns the relationship between cancellation in the monoid associated with a braided set and involutivity of the Yang–Baxter solution. The supplied context records a negative answer to the earlier question of whether 2-cancellativity alone implies cancellativity, but does not indicate a resolution of this exact if-and-only-if formulation.