The eigenvalue conjecture for link Racah matrices

Let $L=(\mathcal{\hat{R}}_1,\mathcal{\hat{R}}_2,\mathcal{\hat{R}}_3)$ and $L'=(\mathcal{\hat{R}}'_1,\mathcal{\hat{R}}'_2,\mathcal{\hat{R}}'_3)$ be triples of $\mathcal{\hat{R}}$-matrices, with $\mathcal{\hat{R}}_i\sim\mathcal{\hat{R}}'_i$ meaning that the two matrices have the same set of normalized eigenvalues, and with $L\sim L'$ meaning $\mathcal{\hat{R}}_i\sim\mathcal{\hat{R}}'_i$ for every $i$. Consider corresponding Racah matrices with the same domain, in bases in which the $\hat{\mathcal{R}}$-matrices are diagonal. Link eigenvalue conjecture. Given two equivalent lists of $\mathcal{\hat{R}}$-matrices $L\sim L'$, the corresponding Racah matrices are equal in these bases. The link case extends the eigenvalue hypothesis from the knot setting to the three distinct eigenvalue sets occurring among the twelve $\mathcal{\hat{R}}$-matrices in the link Yang–Baxter relations; the source does not establish the conjecture in general.