Gaudin-type determinantal norm conjecture for periodic Macdonald spherical functions

Let $R_0$ be a concrete root system, let $c\in\mathbb{N}_{>1}$, and let $P^+_c$ and $P^{\vee,+}_c$ be the corresponding finite sets of dominant weights and coweights. For $\mu\in P^{\vee,+}_c$, let $\Phi_{\xi_\mu}$ be the periodic Macdonald spherical function, viewed as an element of the Hilbert space $l^2(P^+_c,\Delta)$, with quadratic form

$$\langle \Phi_{\xi_\mu},\Phi_{\xi_\mu}\rangle_\Delta=\sum_{\lambda\in P^+_c}|\Phi_{\xi_\mu}(\lambda)|^2\Delta_\lambda.$$

Gaudin-type determinantal norm conjecture. For any $\mu\in P^{\vee,+}_c$, this quadratic norm is

$$\langle \Phi_{\xi_\mu},\Phi_{\xi_\mu}\rangle_\Delta=\operatorname{Ind}(R_0)C(\xi_\mu)C(-\xi_\mu)\det\mathcal{H}(\xi_\mu),$$

where $\operatorname{Ind}(R_0):=|\Omega|$, $C(\cdot)$ is the function defined in the paper's equation (cfun), and $\det\mathcal{H}(\cdot)$ is the determinant of the Hessian defined in equation (hessian). The formula is motivated by the Gaudin determinant paradigm for quadratic norms of Bethe eigenfunctions, but the supplied text does not establish it or state a resolution; its status is therefore open.