In the ODE/IM correspondence for a simple Lie algebra g constructed by Masoero, Raimondo, and Valeri a certain family of meromorphic connections plays a crucial role. A special member of this family of connections turns out to be closely related to the Witten-Kontsevich point in the phase space of the Drinfeld-Sokolov integrable hierarchy associated to g. For this special connection we describe an interesting Langlands duality of Segal-Sugawara operators.