The eigenvalue conjecture for Racah matrices

Let $R$ be a representation, and consider Racah matrices obtained from the tensor product $R\otimes R\otimes R$. Normalize each corresponding $\hat{\mathcal{R}}$-matrix so that the product of its eigenvalues is $1$, and use bases in which the $\hat{\mathcal{R}}$-matrices are diagonal. Eigenvalue conjecture. Given two equal sets of normalized eigenvalues of two $\hat{\mathcal{R}}$-matrices acting on $R\otimes R\otimes R$, the corresponding Racah matrices are equal in their respective bases. This hypothesis asserts that the Racah matrix is determined by the normalized $\hat{\mathcal{R}}$-matrix eigenvalues; the source presents it as a generalization of the corresponding $U_q(sl_2)$ phenomenon, while analogous higher-rank symmetries and fully general analytic formulas remain to be discovered.