Counting admissible polynomial Bethe-root solutions for diagonal and generic parameters

Let $M=2s$, and let $P_a^M(U_i,\bar U_i)=0$ be the polynomial Bethe-root equations for parameter choice $a=d$ or $a=g$. A solution is admissible if its coordinates are pairwise distinct. Admissible-solution counting conjecture. The system admits $2s+1$ distinct admissible solutions. Numerical examples for $M=1,2,3$ support the count, which agrees with the dimension of the relevant vector space; the source gives no proof for generic $M$.