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Goulden and Jackson (1996) introduced, using Jack symmetric functions, somemultivariate generating series $\psi(x, y, z; 1, 1+\beta)$ with an additionalparameter $\beta$ that may be interpreted as a continuous deformation of therooted bipartite maps generating series. Indeed, it has the property that for$\beta \in \{0,1\}$, it specializes to the rooted, orientable (general, i.e.orientable or not, respectively) bipartite maps generating series. They madethe following conjecture: coefficients of $\psi$ are polynomials in $\beta$with positive integer coefficients that can be written as a multivariategenerating series of rooted, general bipartite maps, where the exponent of$\beta$ is an integer-valued statistic that in some sense "measures thenon-orientability" of the corresponding bipartite map.  We show that except for two special values of $\beta = 0,1$ for which thecombinatorial interpretation of the coefficients of $\psi$ is known, thereexists a third special value $\beta = -1$ for which the coefficients of $\psi$indexed by two partitions $\mu,\nu$, and one partition with only one part aregiven by rooted, orientable bipartite maps with arbitrary face degrees andblack/white vertex degrees given by $\mu$/$\nu$, respectively. We show thatthis evaluation corresponds, up to a sign, to a top-degree part of thecoefficients of $\psi$. As a consequence, we introduce a collection ofinteger-valued statistics of maps $(\eta)$ such that the top-degree of themultivariate generating series of rooted bipartite maps with only one face(called unicellular) with respect to $\eta$ gives the top-degree of theappropriate coefficients of $\psi$. Finally, we show that the $b$-conjectureholds true for all rooted, unicellular bipartite maps of genus at most $2$.

Top degree part in $b$-conjecture for unicellular bipartite maps