Cyclotomic Gaudin algebra spectrum conjecture

Let $\lambda_0\in\mathfrak{h}^{*,\nu}$ be the dominant $\nu$-invariant weight satisfying $\langle\check\alpha_i,\lambda_0\rangle\in\mathbb{Q}$ for all $i\in I$. Let $\operatorname{Op}_{{}^L\mathfrak{g}}^\Gamma(\mathbb{P}^1)^{\mathrm{RS}}_{\mathbf z;\lambda_0}$ be the space of cyclotomic ${}^L\mathfrak{g}$-opers with regular singularities at $\mathbf z=\{0,z_1,\ldots,z_N,\infty\}$, residue at the origin given by the finite ${}^L\mathfrak{g}^{\nu}$-oper $[\lambda_0]_{W^\nu}$, and monodromy $e^{2\pi i\lambda_0}$ at the origin. Cyclotomic Gaudin spectrum conjecture. The cyclotomic Gaudin algebra $\mathscr Z^\Gamma_{(z_i)}(\mathfrak{g})$ is isomorphic to the algebra of functions on this oper space; equivalently,

$$\operatorname{Spec}\mathscr Z^\Gamma_{(z_i)}(\mathfrak{g})\simeq\operatorname{Op}_{{}^L\mathfrak{g}}^\Gamma(\mathbb{P}^1)^{\mathrm{RS}}_{\mathbf z;\lambda_0}.$$

This is the paper's second cyclotomic spectral conjecture, identifying the maximal spectrum of the cyclotomic Gaudin algebra with a space of regular-singular cyclotomic opers; its status is not resolved in the supplied text.