Generalized positivity conjecture for matching and constellation coefficients

Let $k\geq 1$ and let $\lambda,\mu^0,\ldots,\mu^k$ be partitions of size $n\geq 1$. Define $c^\lambda_{\mu^0,\ldots,\mu^k}(b)$ and $h^\lambda_{\mu^0,\ldots,\mu^k}(b)$ by the coefficients of $\tau_b^{(k)}$ and $\Psi_b^{(k)}$, respectively, in the generating-series expansions given in the paper. Generalized positivity conjecture. The coefficients $c^\lambda_{\mu^0,\ldots,\mu^k}(b)$ and $h^\lambda_{\mu^0,\ldots,\mu^k}(b)$ are polynomials in $b$ with non-negative integer coefficients. For $k=1$, this extends the positivity conjecture of Goulden and Jackson; the conjecture has been tested computationally in several small cases, but remains open in general.