The uniqueness conjecture for solutions of the $W_r$ monodromy system

Let $W_r$ be the polynomial defined from the unique exponential-Gaudin solution indexed by $r$, and consider the monodromy system in the variables $w_i$ displayed in the source. $W_r$-solution conjecture. The zeros of $W_r$ are the only solutions of that system. The paper suggests that methods for monodromy-free harmonic-oscillator operators may prove this, but it remains conjectural in the supplied text.