  We consider the Dubrovin--Frobenius manifold of rank $2$ whose genus
expansion at a special point controls the enumeration of a higher genera
generalization of the Catalan numbers, or, equivalently, the enumeration of
maps on surfaces, ribbon graphs, Grothendieck's dessins d'enfants, strictly
monotone Hurwitz numbers, or lattice points in the moduli spaces of curves.
Liu, Zhang, and Zhou conjectured that the full partition function of this
Dubrovin--Frobenius manifold is a tau-function of the extended nonlinear
Schr\"odinger hierarchy, an extension of a particular rational reduction of the
Kadomtsev--Petviashvili hierarchy. We prove a version of their conjecture
specializing the Givental--Milanov method that allows to construct the Hirota
quadratic equations for the partition function, and then deriving from them the
Lax representation.
