  Goulden and Jackson (1996) introduced, using Jack symmetric functions, some
multivariate generating series $\psi(x, y, z; t, 1+\beta)$ that might be
interpreted as a continuous deformation of the generating series of rooted
hypermaps. They made the following conjecture: the coefficients of $\psi(x, y,
z; t, 1+\beta)$ in the power-sum basis are polynomials in $\beta$ with
nonnegative integer coefficients (by construction, these coefficients are
rational functions in $\beta$).
  We prove partially this conjecture, nowadays called $b$-conjecture, by
showing that coefficients of $\psi(x, y, z; t, 1+ \beta)$ are polynomials in
$\beta$ with rational coefficients. A key step of the proof is a strong
factorization property of Jack polynomials when the Jack-deformation parameter
$\alpha$ tends to $0$, that may be of independent interest.
