The eigenvalue conjecture for symmetric U_q(sl_N) Racah matrices

Let $U=U\left[\begin{matrix}R_1&R_2\R_3&R_4\end{matrix}\right]$ and $\widetilde U=U\left[\begin{matrix}\widetilde R_1&\widetilde R_2\widetilde R_3&\widetilde R_4\end{matrix}\right]$ be Racah matrices with symmetric representations $R_i$ and $\widetilde R_i$, $i=1,2,3$, specified by the source's tensor-product parametrization and compatibility conditions. Let $d$, $\delta_X$, $\delta_Y$, and $\delta_Z$ be the associated parameters. Symmetric-representation eigenvalue conjecture. If

$$d=\widetilde d,\qquad \delta_X=\widetilde{\delta}_X,\qquad \delta_Y=\widetilde{\delta}_Y,\qquad \delta_Z=\widetilde{\delta}_Z,$$

then the corresponding Racah matrices $U$ and $\widetilde U$ are equal. This is a concrete symmetric-representation form of the eigenvalue conjecture for $U_q(sl_N)$; the cited parametrization and referenced equations are not included in the supplied context, and the general higher-rank problem is presented as nontrivial.