Mukhin–Varchenko conjecture that the norm constant is one

Let $V_{\boldsymbol{\Lambda}}=V_{\boldsymbol{\Lambda}^1}\otimes\dots\otimes V_{\boldsymbol{\Lambda}^k}$ be the tensor product of the relevant $\mathfrak g$-modules. Let $z^0$ have distinct coordinates, let $t^0$ be a non-degenerate critical point of the master function $\Phi(\cdot,z^0,\boldsymbol{\Lambda},\boldsymbol l)$, and let $t=t(z)$ be its locally continued non-degenerate critical point. Denote by $\omega(t(z),z)\in\operatorname{Sing}V_{\boldsymbol{\Lambda}}[\boldsymbol{\Lambda}-\alpha(\boldsymbol l)]$ the corresponding Bethe vector, and let $S$ be the tensor Shapovalov form. The norm formula gives

$$S(\omega(t(z),z),\omega(t(z),z))=C\,\operatorname{Hess}_t\log\Phi(t(z),z,\boldsymbol{\Lambda},\boldsymbol l),$$

where $C$ is independent of $z$. Norm-constant conjecture. The constant $C$ in this formula is equal to $1$. The conjecture specifies the normalization in the norm–Hessian identity for Bethe vectors. The cited theorem establishes the identity with an unspecified constant, while this statement proposes its exact value.