Feigin–Frenkel ODE/IM correspondence for higher quantum KdV states

Let a quantum $\widehat{\mathfrak{g}}$-KdV oper be an oper of the form

$$\mathcal{L}(z,\lambda)=\partial_z+\frac{r-\rho^\vee+f}{z}+(1+\lambda z^{-\hat{k}})e_{\theta}+\sum_{j\in J}\frac{-\theta^\vee+X(j)}{z-w_j},$$

with the additional regular singularities chosen to have trivial monodromy for every value of $\lambda$. Let $Q$ functions be the functions obtained from the connection matrix between the regular singularity at $0$ and the irregular singularity at $\infty$, and let $N$ denote both the level of a state and the cardinality of the set $J$ when these are identified. Feigin–Frenkel ODE/IM conjecture. To any state of the quantum $\widehat{\mathfrak{g}}$-KdV model there corresponds a unique quantum $\widehat{\mathfrak{g}}$-KdV oper whose $Q$ functions coincide with the solution of the Bethe Ansatz equations of the given state. Moreover, the level $N\in\mathbb{N}$ of a state coincides with the cardinality $N$ of the set $J$ of additional singularities of the corresponding oper; in particular, the ground state corresponds to $J=\emptyset$. This is the complete Feigin–Frenkel ODE/IM correspondence for simply-laced $\mathfrak{g}$, identifying quantum KdV states with higher-state opers and their Bethe Ansatz data; the statement is presented as the conjectural extension of the ground-state correspondence.