  Goulden and Jackson (1996) introduced, using Jack symmetric functions, some
multivariate generating series $\psi(x, y, z; 1, 1+\beta)$ with an additional
parameter $\beta$ that may be interpreted as a continuous deformation of the
rooted 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 made
the following conjecture: coefficients of $\psi$ are polynomials in $\beta$
with positive integer coefficients that can be written as a multivariate
generating series of rooted, general bipartite maps, where the exponent of
$\beta$ is an integer-valued statistic that in some sense "measures the
non-orientability" of the corresponding bipartite map.
  We show that except for two special values of $\beta = 0,1$ for which the
combinatorial interpretation of the coefficients of $\psi$ is known, there
exists 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 are
given by rooted, orientable bipartite maps with arbitrary face degrees and
black/white vertex degrees given by $\mu$/$\nu$, respectively. We show that
this evaluation corresponds, up to a sign, to a top-degree part of the
coefficients of $\psi$. As a consequence, we introduce a collection of
integer-valued statistics of maps $(\eta)$ such that the top-degree of the
multivariate generating series of rooted bipartite maps with only one face
(called unicellular) with respect to $\eta$ gives the top-degree of the
appropriate coefficients of $\psi$. Finally, we show that the $b$-conjecture
holds true for all rooted, unicellular bipartite maps of genus at most $2$.
