Summary: In [Trans. Am. Math. Soc. 348, No. 3, 873–892 (1996; Zbl 0858.05097)], I. P. Goulden and D. M. 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 Matchings-Jack Conjecture remains open. M. Dołęga and V. Féray [Duke Math. J. 165, No. 7, 1193–1282 (2016; Zbl 1338.60017); Trans. Am. Math. Soc. 369, No. 12, 9015–9039 (2017; Zbl 1371.05310)] showed the polynomiality of connection coefficients \(c^\lambda_{\pi,\sigma}\) and gave an upper bound on the degrees. We show a dual approach to this problem and investigate Jack characters and their connection coefficients. 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 [loc. cit.].