Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy

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