Exponential enumeration conjecture for set-theoretic Yang–Baxter solutions

A set-theoretic solution of the Yang–Baxter equation is a solution considered up to isomorphism, and its size is the cardinality of its underlying set. Let $n$ denote the size of the solution. Solution enumeration conjecture. There are

$$2^{O(n^2)}$$

solutions of size $n$, up to isomorphism.

The paper compares the total number of solutions with the number of multipermutation solutions of level $2$, for which it proves an upper bound of $2^{(1/4+o(1))n^2}$. The conjecture proposes that the total number of solutions remains bounded by an exponential of order $n^2$, although the source's supplied context does not specify whether the solutions are restricted to a particular class beyond the surrounding discussion.