  In the present paper we discuss the eigenvalue conjecture, suggested in 2012,
in the particular case of $U_q(sl_2)$. The eigenvalue conjecture provides a
certain symmetry for Racah coefficients and we prove that \textbf{the
eigenvalue conjecture is provided by the Regge symmetry} for $U_q(sl_2)$, when
three representations coincide. This in perspective provides us a kind of
generalization of the Regge symmetry to arbitrary $U_q(sl_N)$.
