The Bethe-vector length conjecture

Let $V_1,V_2$ be representations of a simple Lie algebra, let $X$ be a Bethe vector obtained from a critical point $t^0$ of the corresponding phase function $\Phi$, and let $B$ be the tensor-product Shapovalov bilinear form. Set $\kappa=1$ in $\Phi$. Bethe-vector length conjecture. The length of $X$ is the Hessian of the logarithm of $\Phi$ at $t^0$:

$$B(X,X)=\det\left(\frac{\partial^2}{\partial t_i\partial t_j}\ln\Phi(t^0)\right).$$

The claim relates the Shapovalov norm of a Bethe vector to the local Hessian at the corresponding critical point; the supplied text gives no resolution status or general result beyond the preceding discussion of uniqueness.