  Using Jack polynomials, Goulden and Jackson have introduced a one parameter
deformation $\tau_b$ of the generating series of bipartite maps, which
generalizes the partition function of $\beta$-ensembles of random matrices. The
Matching-Jack conjecture suggests that the coefficients $c^\lambda_{\mu,\nu}$
of the function $\tau_b$ in the power-sum basis are non-negative integer
polynomials in the deformation parameter $b$. Do{\l}\k{e}ga and F\'eray have
proved in 2016 the "polynomiality" part in the Matching-Jack conjecture, namely
that coefficients $c^\lambda_{\mu,\nu}$ are in $\mathbb{Q}[b]$. In this paper,
we prove the "integrality" part, i.e that the coefficients
$c^\lambda_{\mu,\nu}$ are in $\mathbb{Z}[b]$.
  The proof is based on a recent work of the author that deduces the
Matching-Jack conjecture for marginal sums from an analog result for the
$b$-conjecture, established in 2020 by Chapuy and Do{\l}\k{e}ga. A key step in
the proof involves a new connection with the graded Farahat-Higman algebra.
