$Q$-functions for lambda opers

We consider the Schrödinger operators which are constructed from the $\lambda$-opers corresponding to solutions of the ${\widehat{\mathfrak{s}\mathfrak{l}}}_2$ Gaudin Bethe Ansatz equations. We define and study the connection coefficients called the $Q$-functions. We conjecture that the $Q$-functions obtained from the $\lambda$-opers coincide with the $Q$-functions of the Bazhanov-Lukyanov-Zamolodchikov opers with the monster potential related to the quantum KdV flows. We give supporting evidence for this conjecture.