Notions of equivalence among constant Yang--Baxter Operators: To $\mathrm{\infty }$-Equivalence and Beyond

Recent advances in the classification of certain classes of Yang-Baxter operators through a holistic categorical approach has some of us wondering about ways to identify solutions. In particular the classification of charge conserving solutions in all dimensions inspires the question: \emph{what fraction of solutions are $\mathbf{\text{close}}\mathbf{\text{to}}$ charge conserving solutions, under any $\mathbf{\text{reasonable}}$ notion of equivalence?} Of course \emph{reasonable} and \emph{close to} are open to interpretation. Here we suggest some precise notions of both, and prove that in dimension $2$, 10 out of 11 local families of solutions are close to charge conserving, and the one that misses is a singleton.