The uniqueness conjecture for critical points of Bethe phase functions

Let $\mathfrak{g}$ be a simple Lie algebra, let $V_1,V_2$ be representations of $\mathfrak{g}$, and let $\Phi$ be the corresponding phase function in variables assigned to the simple roots. Critical points satisfy $d\Phi=0$, and points related by permutations of variables assigned to the same simple root are identified. Uniqueness conjecture. If the space of singular vectors of a given weight in $V_1\otimes V_2$ is one-dimensional, then the corresponding phase function has exactly one critical point modulo permutations of variables assigned to the same simple root. This holds for $\mathfrak{g}=sl(2)$ by the stated symmetric-function description, while the general case is left open.