  In 1996 Goulden and Jackson introduced a family of coefficients $( c_{\pi,
\sigma}^{\lambda} ) $ indexed by triples of partitions which arise in the power
sum expansion of some Cauchy sum for Jack symmetric functions $( J^{(\alpha
)}_\pi )$. The coefficients $ c_{\pi, \sigma}^{\lambda} $ can be viewed as an
interpolation between the structure constants of the class algebra and the
double coset algebra. Goulden and Jackson suggested that the coefficients $
c_{\pi, \sigma}^{\lambda} $ are polynomials in the variable $\beta := \alpha-1$
with non-negative integer coefficients and that there is a combinatorics of
matching hidden behind them. This \emph{Matchings-Jack Conjecture} remains
open. Do\l{}\oldk{e}ga and F\'eray showed the polynomiality of connection
coefficients $c^\lambda_{\pi,\sigma}$ and gave the upper bound on the degrees.
We give a necessary and sufficient condition for the polynomial $ c_{\pi,
\sigma}^{\lambda}$ to achieve this bound. We show that the leading coefficient
of $ c_{\pi, \sigma}^{\lambda}$ is a positive integer and we present it in the
context of Matchings-Jack Conjecture of Goulden and Jackson.
