Equivalence conjecture for the Kirk–Livingston–Wang and quotient string-link representations

Let $\mathrm{pStr}(n)$ be the monoid of pure string links. The Kirk–Livingston–Wang representation is

$$\gamma^{KLW}_n:\mathrm{pStr}(n)\to\operatorname{End}\bigl(\mathbb{Q}(t_1,\ldots,t_n)^n\bigr),$$

and let $p\mathcal W_{n,1/0}$ denote the corresponding quotient of the two-variable R-matrix representation. Equivalence conjecture. The representation $\gamma^{KLW}_n$ is equivalent to $p\mathcal W_{n,1/0}$. This conjecture extends the preceding theorem, and the authors expect its proof to follow a similar strategy.