The Macdonald generalization of the Matchings-Jack conjecture

For partitions $\pi,\mu,\nu$ of the same size, let $\boldsymbol{c}^\pi_{\mu,\nu}(\alpha,\gamma)$ be defined by

$$\sum_{\lambda\in\mathbb{Y}}u^{|\lambda|}t^{-2n(\lambda)}q^{n(\lambda')}\frac{{\mathfrak J}^{(\alpha,\gamma)}_\lambda[X]{\mathfrak J}^{(\alpha,\gamma)}_\lambda[Y]{\mathfrak J}^{(\alpha,\gamma)}_\lambda[Z]}{\widetilde{j}^{(\alpha,\gamma)}_\lambda}=\sum_{m\geq0}\sum_{\pi,\mu,\nu\vdash m}\frac{u^m\boldsymbol{c}^\pi_{\mu,\nu}(\alpha,\gamma)}{z_\pi(q,t)}p_\pi[X]p_\mu[Y]p_\nu[Z],$$

where $\widetilde{j}^{(\alpha,\gamma)}_\lambda=\gamma^{-2|\lambda|}j^{(q,t)}_\lambda=\langle{\mathfrak J}^{(\alpha,\gamma)}_\lambda,{\mathfrak J}^{(\alpha,\gamma)}_\lambda\rangle_{q,t}$, and $b:=\alpha-1$.

The Macdonald generalization of the Matchings-Jack conjecture. For any positive integer $n$ and partitions $\pi,\mu,\nu$ of $n$,

$$(1+\gamma)^{n(n-1)}z_\mu z_\nu\boldsymbol{c}^\pi_{\mu,\nu}(\alpha,\gamma)$$

is a polynomial in $b$ and $\gamma$ with non-negative integer coefficients. This is a Macdonald deformation of the Jack Matchings-Jack positivity conjecture, and the supplied text gives no resolution evidence.