Uniqueness conjecture for the spectral parameters of arbitrary-rank quantum orbits

Let $p$ be the rank parameter and consider the system

$$\prod_{i\ne j}\frac{q\mu_j(\lambda)-q^{-1}\mu_i(\lambda)}{\mu_j(\lambda)-\mu_i(\lambda)}=\prod_{i\ne j}\frac{(\lambda_{p-i+1}-\lambda_{p-j+1}+i-j+1)_q}{(\lambda_{p-i+1}-\lambda_{p-j+1}+i-j)_q},\qquad 1\leq j\leq p.$$

A solution is given by

$$\mu_i(\lambda)=\eta(\lambda)q^{-2(\lambda_{p-i+1}+i)},$$

where $\eta(\lambda)$ is an arbitrary nonzero multiplier.

Uniqueness conjecture. At an arbitrary value of $p$, this solution of the system is unique.

The text establishes uniqueness for $p=2$ because the system is then linear in the $\mu_i(\lambda)$, but gives no resolution for arbitrary $p$.